It seems interesting that the term "loop" is used in both topology and abstract algebra. Is that just a coincidence, or is there a connection between them?
Everything I have seen in my research associates the topological idea of loop more to groups than abstract algebra loops.
The choice of word is probably just a coincidence. I haven't heard of any deep connection, nor can I spot one.
That is not to say that you can put a (algebra)loop structure on a set of (topology)loops. As you say, there are a few natural choices of operations between (topology)loops but those most often result in full-fledged groups.
Naming collisions like this happen all the time, of course. Consider "domain" and "regular" etc.