I've been wanting to start learning about Polish spaces because of their relationship with classification theorems.
Polish Space: A space homeomorphic to a seperable complete metric space; source
However the notion of morphisms and importantly of equivalence of Polish spaces is missing from both nLab and Wikipedia. It's unclear to me what constitutes as equivalence of Polish Spaces from these two articles, so my question, Is there a standard notion of equivalence of polish spaces?
A polish space is a topological space that happens to be separable and completely metrizable. The only natural choice of arrow should be continuous maps between them, which makes homeomorphism the correct notion of isomorphism.
This is also backed up in the literature. For example, in Kechris's Classical Descriptive Set Theory, the notion of homeomorphism is used constantly to compare polish spaces. See the famous theorem that every polish space is homeomorphic to a closed subspace of $\mathbb{R}^\mathbb{N}$ (I.4.17 in Kechris).
Since you mention "measurable bijection", it sounds like you're interested in the borel structure of a polish space. If we forget about the topology and only remember the borel sets, then we get a standard borel space. Here the natural choice of arrows are the measurable functions, and thus our isomorphisms will be measurable bijections with measurable inverses.
Again, this is reflected in the way standard borel spaces are actually used. See the notion of "borel isomorphism" defined in theorem II.14.12 in Kechris, which is then used throughout the rest of the book.
I hope this helps ^_^