Does it make sense to let the class implement an interface instead of using an anonymous inner class - in regards to object allocation?

Yes. It saves one allocation. If the class is private or internal and is not part of some widely-used API, then it makes sense.

Why doesn't it make sense on a widely used API?

If the class is a part of public API and it implements some listeners interface, then it leaks its implementation details (and what happens if client just invokes a method of that listener interface?)

I'm only taking about memory / performance, not API design