Couldn't this translate to the literal "you have animals?"
Sometimes in English, you need the "do", sometimes it's incorrect. Take the following two examples:
Person A: I have animals. Person B: You have animals?! -- "Do" would never be said.
Person A: I'm going to open a zoo. Person B: Do you have animals? -- No "do" would be improper/colloquial.
But since Portuguese doesn't have the "do you" vs "you" distinction, it's not clear just from "tu tens animais" whether the Portuguese speaker intentionally meant to include it or leave it out. Doesn't it make sense given the lack of specificity to accept either?
The word pet, can be translated that way in some contexts. The best way to say "pet" in Portuguese is: "animal/bichinho de estimação". As I've said in the beginning you can say "animal" for pet, but that is when the ideia "de estimação" is already in there, so you use only the first word to mean the whole translation.
At one time it was considered to be American usage, but It is now standard in BrE. M. Swan's "Practical English Usage" (OUP) describes modern British English in this quotation:
"In British English, short question and negative forms of have are possible though they are often formal.
• Have you an appointment? (formal GB only)
• Do you have an appointment? (US/GB)"
ngrams BrE (2009) https://tinyurl.com/y8nfbunt