Can the number of tables and functions affect performance?
It's a gradual process and the effect is comparatively small. But of course, looking up entries in system tables gets slower with lots of rows. Those are just regular tables. Highly optimized, but regular tables. There are indexes to keep the effect small.

I have never actually noticed an effect in my biggest database with a couple of hundred tables and close to 1000 functions. Sequences, views, triggers .. everything. Your DB seems to have similar numbers - I'd guess you cannot report any observations yourself? Or you would surely have added that in your question.

The cost for other things like backup and VACUUM rises, too, obviously.