Note in advance: I am assuming that the abstract realm you're referring to is something along the lines of the "third realm" that Frege posited (alongside physical and mental domains).
Philosophers vs. early set theorists: these are not really very distinct categories. To some extent, we can distinguish between a trend in analytic philosophy and a trend in early, mainline set theory, though, one with expansionist aspects and one with restrictionist ones. This will be my core point of departure in what I write from here.
From Alexus Meinong to David Lewis: so first note that Alexus Meinong, in the apparent camp of the analytic philosophers, did have a rather unrestricted abstract realm to his viewpoint's name. As they say:
Not only existing things but also all kinds of non-beings (nonentities, Routley 1980: 7) find their place in Meinong’s all-embracing theory of objects — among them even impossible objects, like the round square, as well as paradoxical, “defective” objects, like some forms of the liar [the liar sentence] or special cases of purely self-referential thinking (the thought about itself, for example).
Meinongianism has reemerged and evolved into modern talk of "impossible worlds", which are supposed to be yet possible and actual (as worlds, though their inhabitants are not possible). But the most jarring purported impossibilia, i.e. true contradictions, are not admitted so much among plenitudinous Platonists at least so far as it is said:
If every consistent mathematical theory is true of some universe of mathematical objects [emphasis added], then mathematical knowledge will, in some sense, be easy to obtain: provided that our mathematical theories are consistent, they are guaranteed to be true of some universe of mathematical objects.
One version of this kind of Platonism preserves Meinong's round square, without meaning to actualize a contradiction, by having the round square as an object that encodes incompatible properties without exemplifying them: the round square is not actually round or square, much less both at once.
A rather different doctrine in the same extended family (the family of "plenitudinous" theories, in this case a plenitude of concreta), David Lewis' modal realism, was cited by Brian Greene (the string-theory popularizer) as a philosophical background for the string-theoretic landscape of zillions of "universes," but whereas Lewis estimated the number of modal-realist universes to be among the transfinite cardinals, Greene's string theory "merely" involves approximately 10500 universes. But so even Lewis' multiverse seems fairly restricted (the specific cardinal he guessed at is usually calculated to be radically infinitesimal when compared to the bounty of higher set theory). Now, the Lewisian multiverse is, again, a case of concreta, except insofar as one might interlock Edward Zalta's encoding-vs.-exemplifying doctrine of abstracta with Lewis' indexical theory of actuality (so that we have an indexical theory of abstracta). But even aside from considerations of such a fused theory, we might style the relatively nonactual Lewisian worlds as functionally equivalent to abstracta from within any given such world. At any rate, generic possible worlds (conceptually disjoint over the bare Kripkean sense of these things or hypostasized in a Lewisian manner) and their ensemble aren't seemingly so restricted; and then impossible worlds are even less so.
Why were many early set theorists concerned to prove the well-ordering principle? Recall that Cantor did not believe that "all the (particular) sets" were contained in some general set, but he thought that the divine intellect (which he did understand in terms very much like Kant's talk of intellectual intuition) was what they were all elements of, and this intellect was explicitly stated to be a particular (the individual personality of the divine nature).
But, parallel to Frege and ever since, most set theorists who have spoken of a universal set or universal class or other such notions, have represented this entity rather impersonally (though note the perpetual talk of "witnesses" who "think" various things in set theory, to this day). More pointedly, there was, early on, a major push to establish the well-ordering principle. Why?
By itself, Cantor's original diagonal argument about the lack of a bijection between ℕ and ℝ only showed just that: a difference in "size," not a determinate relation of lesser-and-greater. Cantor separately came up with the series of the ℵa (which essentially does obey a uniform lesser-or-greater commandment) and then advanced the Continuum Hypothesis, but it was an open question whether |ℝ| would be one of the ℵa. Were it not, then transfinite arithmetic might've been even easier to represent than if (G)CH were true; the expression "2ℵ0" would just go to some independent symbolism, "κ0" say, and then "2κ0" could've just gone to "κ1," and all the κb would be their own series, and we would have not detained ourselves with the intricacies of trying to overlap the aleph and kappa numbers.
But oftentimes the emotionally-minded reason for mathematicians to "invent" a new kind of number is to use it in calculations that connect it to other kinds of numbers. It was easy enough (eventually) to get the imaginary unit and use it to keep the symbolism for real numbers with the symbolism for complex numbers, and so to facilitate the arithmetic for going back and forth from ℝ-terms solely to ℂ-terms inclusively. It took a lot longer to do something similar with infinitesimals (I think we had to wait on Abraham Robinson's nonstandard analysis for this, and NA is still not as symbolically "friendly" as John Conway's surreal infinitesimals), but we still did it. So for transfinite arithmeticians, the desire was to keep the terms of their arithmetic connected up enough so that the successor operation on the zeroth aleph, which accesses ℵ1, would "be close friends with" the powerset operation which accesses |ℝ|. But then we had to "prove" the well-ordering principle, so that the series of powersets and the series of successors, among the alephs, would involve commensurable terms, and we could definitively write expressions like, "ℵ1 ≤ |ℝ|."
The introduction of the axiom of foundation had to do with Russell's paradox, albeit not absolutely directly. If all sets are not elements of themselves, and this due to an axiom as such, the very question of a set that would be and not be an element of itself is not quite so "askable" in the first place, and the paradox is dismissed on the level of preliminary semantics. However, the relationship between the foundation axiom and the Russell paradox is thematic more than expressive of a technical requirement, so though restricting the abstract realm of mainline set theory to well-founded sets was par for the course for years on end, Quine and then Aczel loosened the restriction on parafounded sets among mainline set theorists. And now, especially in the light of the Hamkins multiverse, we have broken those chains and melted many of them down.
Now, the foundation axiom is separate from the well-ordering principle. You can have sets that are well-ordered but circular (or which head up infinite descending elementhood sequences) as well as noncircular/nondescending but not well-ordered. Still, I suspect that whatever feeling we have for transfinite arithmetic (at least in the sense of the desire that there be a nifty system/method of this arithmetic) depends very much on both principles being in play. A series of parafounded alephs would be hard to commensurate (maybe impossible to commensurate) with the well-founded ones in the tidy way envisioned by Cantor. It would be like ending up with the κb all over again, maybe.
Yet so when Paul Cohen proved that ZFC cannot decide which aleph the Continuum's cardinality is, he was able to undermine the going-back-to-Cantor hope for a reliable/stable system of transfinite arithmetic. Two years later, Paul Benacerraf identified the identification problem; roughly around the same time, Robinson devised nonstandard analysis; and so attitudes and viewpoints like game-theoretic formalism and general pluralism were on the up and up again. The now-prevailing conception of set-theoretic multiverses is a manifestation of analytic-philosophical ensembles of possible and impossible (and transpossible) worlds in the sense that Hamkins, Gitman, and so on are the inheritors of plenitudinous Platonism in a way that is (sociologically speaking) very much like how Brian Greene inherited much of his motivation to believe in the string-theoretic multiverse from another plenitude-theoretic analytic philosopher (Lewis).
Conclusion: so, among the community of analytical philosophers (from Frege and Meinong onward), there was a less insistent demand that the abstract realm be subject to complicated restrictions on its content (noncontradiction was not even held to too fast "in the beginning" and is even less mandatory nowadays). For their own peculiar sentimental (aesthetic) reasons, set theorists concerned to develop a system of transfinite arithmetic like unto the arithmetic for finite numbers that we know and love were keen to introduce restrictions into their abstract realms, until the time of Cohen and Benacerraf came and that keening (metaphorically) turned to wailing and gnashing of teeth. Then the less-restricted abstract (and concrete) realms current in the theorizing of analytic philosophers dovetailed with the "universe of sets" more, until now we have that neither family of abstract-realm doctrines (philosophical or mathematical) are so distinguished, but Plato's heaven has married Cantor's paradise, and their children are the multiversal narratives (and metanarratives) popular in physics and fiction nowadays, too.
