you cant "wish a dog" in English. you can only want a dog, or wish for a dog.
a dori:
eu doresc
tu dorești
el/ea dorește
noi dorim
voi doriți
ei/ele doresc
It should be: they desire a dog, I desire a cat
It shouldn't.
Desire is a perfectly valid meaning for the verb "a dori" and considering the OP was made 10 months ago, when it appears that the original translation was given as "they wish a dog, I wish a cat", then yes, their suggestion was correct.
Exactly