When people say:

I been to a lot of different places in the world.

I been pretty good lately.

Is it still considered present perfect tense? Why is the have omitted and is it proper grammar?
Both sentences are wrong. The word "have" (or its contraction in "I've") is required.
Personally I never saw this on paper as well as hearing someone pronouncing it. Maybe you just didn't hear the 've. In this case I've is pronounced "aiv".

It's not a proper word use and isn't considered present perfect. Maybe it's a sort of a particular accent. But it's false.