And slang is a part of a living language.
When we converse, we need to adapt our diction to the situation. Using formal, structured grammar and usage is the right choice in many situations, especially when dealing with business relations, diplomacy, or academia. On the other hand, when speaking or writing in informal, friendly groups, the use of common slang is a good way to show friendliness. In such forums, strict adherence to formal grammar is not considered essential. In fact, it might be considered inappropriate.
It might even be argued that there is nothing wrong, grammatically, with "my bad." Walter chose to mention that "my" shows possession, which puzzles me because the chief grammatical problem with the expression is that "bad" is an adjective that is supposed to describe something, but it has nothing to describe.
Or does it?
If one were to write, "He is taller than I," that sentence could be considered grammatically incorrect. "I" has no formal purpose in the sentence as written, but the sentence is considered correct because part of the sentence is understood. The complete sentence is "He is taller than I am tall." One could argue that in the case of "my bad," the rest of the sentence--including whatever it is that is bad--is also understood.
But arguing that would be pointless. This is a friendly forum in which one's diction should not be a subject for discussion, should it?