I can't understand if "banana" or "a banana" has the same meaning. I can understand that "the" and "a" are for definite and indefinite nouns. But what about those which aren't preceded by either of them?
The phrases definitely have different meanings.
I have eaten a banana.
I have eaten one single banana.
I have eaten banana.
I have eaten something that contained some amount of banana.
Additionally, if you said these to me I would interpret the first one as having been recently while the second one would be at a further point in the past. Like explaining that you have eaten banana before and are familiar with bananas. But that's all implication based on experience and isn't supported by definitions.
You should say 'I have ate a banana' or 'I have eaten banana'. Proper use is that you have ate a banana recently (first case). Second case means you have ate banana sometime in your life. I have eaten a banana is not proper usage unless someone asks you if you have ever ate a banana. This answer would be formal i.e. Yes, I have eaten a banana or yes, I have eaten banana. Eat is the present simple.
Ate is the past simple.
Eaten is the past participle.