You do find a good number of "Café du Cerf"'s, although there are other places such as hotels and restaurants that bear that name; the name might be "Au Cerf" or "Le Cerf".
It is quite similar to Britain, where the word "stag" is used to name not only pubs but also inns and restaurants.
There is another equivalent in France, provided you tolerate a little shift in the type of establishment you are considering; it is not often going to be a café but instead it'll be an inn or hotel (where there is bound to be a bar and/or lounge). The privileged animal, not wild this time but fully domesticated, is the horse, and without exception, the white horse or the black horse. You'll find in all parts of the country the names "Auberge du cheval blanc" and "Hôtel du cheval blanc", "Le cheval Blanc", etc.
You do find in Britain an exact correspondence for this last cultural trait since there the word "horse" is often used in the names of places such as pubs, restaurants and inns, a few of these inns being as well pubs.
There are four main types of names of pubs you could find in France as far as I know.
Names related to some cultural aspect of the city, region you are in (la bonne mère, la tour eiffel...)
British names (Shannon, Shamrock, Black Horse....)
Puns on names including names starting with " O' ". I believe these are so common (restaurants, fast foods, cafes, pubs..) because of the pun. "Au soleil" would be "O' Soleil", and that would imply that your pub has a nice, sunny terrace
Name of someone, probably the owner -> "Chez Marie", "La maison de Ben"...