jumpdeer My question is, how is Snapchat able to do this if they do not have access to unique identifiers?
The name of that FAQ entry is not "unique identifiers"; it's "hardware identifiers". Scrolling down, the next FAQ entry describes potentially-unique non-hardware identifiers. In addition to what's listed there, issue #2314, about the Media DRM i.d., is still open.
I don't know which permissions Snapchat requires, or what information it might acquire from Google Play, but perhaps it has access to phone numbers?
Aside from the issue of potential actually-unique identifiers, large platforms may engage in statistical profiling, i.e., educated guesses about whether a new account is legitimate according to their thinking. They generally aren't interested in disclosing their exact methods.