The answers to both "some kind of platonism" and "infinitely many universes instantiating mathematical structures" is no. Born is closer to Hegel than to Plato, and even further from Tegmark than from Plato.
His views are described more systematically in the book Physics in My Generation (1966), the chapter Symbol and Reality:
"In every field of experiences this correspondence of sense impressions with symbols has been established... I wish to speak only about the exact
sciences which I know. There mathematical symbols are used, and they have a particularity: they reveal structures.
Mathematics is just the detection and investigation of structures of thinking which
lie hidden in the mathematical symbols... These are structures of pure thinking. The transition to reality is made by theoretical physics which correlates symbols to observed phenomena. Where this can be done hidden structures are coordinated to phenomena; these very structures are
regarded by the physicist as the objective reality lying behind the subjective phenomena."
First, the second paragraph indicates that Born, unlike Tegmark, separates physical from "platonic"/mental existence, and has no need to instantiate his structures as physical universes. Second, Born espouses what is now called mathematical structuralism. This does mean that mathematical claims carry objective truth, but it does not mean that what they are about (like structures) exists even platonically. Structuralism is certainly compatible with platonism, but it does not compel it, it is even compatible with some forms of nominalism. The structures can supervene on something else (like actions or communal practices), and the mathematical talk of them can be expressing that in a different guise. For Born, being objective stems from being "communicable, controllable", and is more of a Hegelian than platonic variety (a la Tegmark), perhaps with a Peircean twist, see SEP Peirce's View of the Relationship Between His Own Work and German Idealism. Another parallel is to Poincare's structural realism with a Kantian flavor.
Born himself invokes Kant and Hegel to clarify his position, and even adapts Hegelian take on Kant's thing-in-itself to his own conception. He goes even further than necessitarian Hegel in admitting:
"The experimentalist has the choice which of them to employ. Thus a subjective trend is reintroduced into physics and cannot be eliminated. Another loss of objectivity is due to the fact that the theory makes only probability predictions".
Nonetheless, the following passages suggest that a purely nominalistic structuralism, where the structures are "empty abstracta", is not for Born either:
[...] The assumption that the coincidence of structures revealed by using different
sense organs and communicable from one individual to the other is accidental, is
improbable to the highest degree... The
concept of causality is a residue of former ways of thinking and is replaced today
by the process of coordination as described before. This procedure leads to structures
which are communicable, controllable, hence objective. It is justified to call these
by the old term 'thing in itself.' They are pure form, void of all sensual qualities.
That is all we can wish and expect...
If the object of modern physics, in particular those of atomic physics, are identified with KANT'S 'thing in itself' one can agree with HEGEL that they are a 'perfect abstractum.' But that they are perfectly empty, something from a world beyond, does not fit the facts. Remember what practical use can be made of them in the production of things like engines, aeroplanes, nuclear reactors, plastics, electronic computers and so on ad infinitum. It might happen that nuclear research leads to our being transfered to 'the other world.' Yet HEGEL did not mean this and could not foresee it.'
As usual, it is disputable what Hegel meant or foresaw (steam engines were already around, and there is nothing, in principle, preventing abstracta from being useful in practice). The kind of thin but real existence of abstracta, along with their supervenience on actions and "habits" as the objectively real patterns in them, is reminiscent of Peirce's "extreme realism", derived from merging Kant and Hegel with Duns Scotus, see What is existence and how far does it extend? Although it is unlikely that Born was familiar with it. The reference in the quote is to Hegel's Encyclopedia of Philosophy, §44:
"The thing in itself... means the object as far as everything referring to consciousness, feeling,
emotion as well as to all notions is abstracted. It is easy to see what is left - the perfect abstractum, the complete emptiness, just something from 'the other world (Jenseits)...'".