The gramatically correct form is "a dog and a cat". You do see/hear "a dog and cat" which has an identical meaning and people will understand it perfectly well.

I feel like the second requires more: I have a dog and a cat. I have a dog and cat that like to play together. I have a dog and cat that always fight. I have a dog and cat that I got from the animal shelter.

I think the sentence " I have a dog and a cat" is correct.