It is important to distinguish between small p platonism -- the belief in the existence of abstract objects, and large P Platonism -- the belief that Plationic Forms are the most fundamental aspect of reality. Per the philosopher surveys -- the majority of philosphers believe in one or another form of small p plationism.
I say one form or another, because small p platonism is just the acceptance of SOME kind of abstract object realism, and whether that is Objective Morality, math realism, logic realism, information realism, function/algorithm realism (functionalism in philosophy of mind is platonic, which Jaegwon Kim points out in his critique of supervenience), acceptance of one does not require acceptance of all.
How one fits platonism into an ontology is also not necessarily uniform for small p platonists. Current physicalists are generally not MONISTS about ontology. See Melnyck: https://www.amazon.com/Physicalist-Manifesto-Thoroughly-Materialism-Philosophy/dp/0521827116/ref=sr_1_1?crid=1HE6Q2L5IUZ12&keywords=physicalist+manifesto&qid=1641751800&sprefix=physicalist+manifesto%2Caps%2C138&sr=8-1, Stoljar: https://www.amazon.com/Physicalism-Problems-Philosophy-Daniel-Stoljar/dp/0415452627/ref=sr_1_3?crid=3ANOGUZE7L2F2&keywords=physicalism&qid=1641751881&sprefix=physicist+%2Caps%2C175&sr=8-3 and Papineau: https://www.researchgate.net/publication/265149272_The_Rise_of_Physicalism
All of these physicalists (and near physicalists -- Stoljar abandoned physicalism as a result of his study of it but has strong near-physicalist inclinations still) admit to the existence of the non-physical, and even agree that the non-physical may be causal on the physical. They just hold that physicalism requires at some point in such a causal history, one can trace a non-physical cause back to a physical cause. This would allow a platonic real to be either emergent from physicalism (hence tracing its causal history to physics), OR prior exiting, but only interactive to the material in a derivative way.
Most math realists today hold that infinite maths exist, and mathematicians discover pre-existing math. This is also a newly developing consensus among logicians, that logic is infinite, and logicians discover it: https://www.cambridge.org/core/journals/think/article/abs/guide-to-logical-pluralism-for-nonlogicians/EDFDFA1C9EB65DB71848DABD6B12D877
The most coherent ontological view that I have seen supporting independent platonism is Frege's 3-world model. Popper also adopted 3 worlds as an ontological view, and provides a good summary of how he saw the ontology working in this lecture; https://tannerlectures.utah.edu/_resources/documents/a-to-z/p/popper80.pdf. Note in Popper's view, consciousness is emergent from matter, but then operates with causal INdependence from its initial substrate. Abstract ideas include ALL of math, logic, information, morality, ideas, memes, and science hypotheses. These all pre-exist, and are discovered by us. Abstractions cannot be interactive with matter directly, and are only interactive through the medium of consciousness.
The SEP entry on abstract objects assumes that Frege and Popper's assumption that abstract objects cannot be causal on matter must be true. Note however that modern physicalists do not require this even of physicalism, and platonism can involve abstract to matter causation. This is particularly true in the practice of theoretical physicists. A popular minority view among such theoreticians is that math actually is what physics reduces to. IE that everything physical actually is mathematics (physicalism therefore reduces to a version of idealism). A less explicitly idealist version of this is the belief that math constrains and infers the physical. This weaker dependence is the reasoning that Sean Carroll gives for accepting the Everett "interpretation" of quantum mechanics - the other universes are there in the math, therefore since ethe math works, he thinks they're actually existing!
This is a long discussion but I hope it explains how small p platonism can work. Both emergence, and pre-existence are possible with it, but emergence is generally for the strata, not each object. Platonic objects are generally seen as discovered, rather than created individually.