Pets are usually chosen depending on the character of the individual who owns them. Animals chosen as pets usually the best suited to their owners specifically. In the Marvel universe, characters are diverse and unique in their way. Some are fast, some intelligent, others are compelling, and even gods in their world. Baring all these factors in mind, we can quickly know what kind of animals each of the Avengers would own.
Tony Stark spends a lot of time in the garage with his tech and needs a low-maintenance pet. He would have a cat as they require little attention and would do well in the mansion. Captain America would have a golden retriever, given he has a classic American lifestyle. Bucky Barnes, who is the Winter Soldier, looks like a guy who enjoys solitude. Life alone in the coldest mountains and own a husky, maybe a pack of them even. Dr. Strange is a magician who could enjoy the company of something mythical like a phoenix.
Thor stated that in the past, he had a liking for snakes, something like a python would therefore do him great good. Falcon would have a falcon as a pet considering his name is a derivative of the bird. Hawkeye obviously would also own a pet that seems to have inspired him, hence a hawk. Loki is a cunning character, the king of deception so that he would hold a fox. Spider-Man is a nerd, to be fair. With this kept in mind, he would, like Tony, have a low maintenance pet like a lizard. War Machine, who is a Military guy, would probably own a loyal German Shepherd.
Black Panther seems more likely to have a tamed panther than any other pet. Captain Marvel has a favorite pet named goose, technically speaking. The Black Widow is ferocious and deadly, a tiger seems to suit a character with such finesse traits. Hulk is relatively intelligent but very dangerous, a gorilla is a good fit for such an enormous creature. Maria Hill already has a pet named India that likes Nick Fury. Scott Lang loves his daughter very much, a hairy dog like a Basset Retriever is ideal.
Okoye is a true warrior and is best fit to have a lioness as her lovely companion. Shuri, who is King T’Challa’s sister, has a spider-monkey named Abu. The vision would have a bird sanctuary while Wanda enjoys a Persian cat.