"You got an issue? I got a problem with that.."

Is the grammar correct? And what does that sentence mean?

No, to me it is very incorrect.

"Do you have an issue? I have a problem with that!"

An issue is an attitude which leads to confrontation and argument. The speaker is accusing the other person of being belligerent and confrontational and in turn is being aggressive.

It should be -

"Do you have an issue with that?" - "I have a problem with that"

or

"Do you have a problem with that?" - "I've got a problem with that"

I think so anyway x

The grammar is o.k.
Meaning;
Do you have a problem[issue]? I have a problem with that.
I prefer my sentences however the others are widely used.
I translated issue asproblem.
Suppose you do not like someone then you have an -issue with them.

For spoken/colloquial English, the grammar is acceptable and common. For written/formal English, it is not very pretty.