You need to hibernate one pet before you can adopt another UNLESS you have a gourd house not in use.
You can have as many pets awake as gourd houses. For example:
I have 3 gourd houses, but only two pets - Tranquile and Steven. When both of them are awake I can still adopt a third pet and chose who is greeter and who is visitor. If I hibernate one pet, I then have two free gourd houses to adopt more pets with.
Say you have two gourd houses and two pets (Dog and Cat).
You hibernate pet Dog.
You now have your active greeter (Cat) and an open gourd house to adopt a new pet (Dragon).
After adopting you can then pick who is greeter and who is visitor (between pets Cat and Dragon), or hibernate pet two of the three pets to have the same pet as greeter and visitor (EG hibernating pets Dog and Cat so pet Dragon is both roles).
To hibernate a pet you will view your pet page, click the "edit" icon, and then scroll down below the welcome section to where your pet is listed. Clicking on the little moon will hibernate your pet.