is は/が always needed to make a grammatically correct sentence?

Question

I listen to a lot of Japanese music, and a common trend I've seen with a lot of them are to say 「≪pronoun≫≪noun≫」. For example,

「あたし猫」 = "I'm a cat."

My question is, is this grammatically correct? I thought you needed は or が to mark the subject/topic, like so:

「あたしは猫」/「あたしが猫」= "I'm a cat."

Which would make the former:

「あたし猫」 = "I, cat."

But, I know Japanese doesn't abide by English grammar standards, nor does は/が inherently mean "is/am"etc. So, I ask,

1. How would「あたし猫」be translated?
2. Is it grammatically correct? (Even if the English translation turns out to be something like "I, cat."?)

Answer 1

1. It means exactly the same as あたしは猫/あたしが猫
2. Grammatically it is incorrect. But in reality(at least for Japanese), most of the sentence we speak is not grammatically correct at all when we are talking with friends.
   Ex) この後何する？