It depends on whose aunt this is, and to whom the speaker is talking. I would say: "We should've complimented Kathy on her coffee set!” ...because I am familiar with my aunt and call her by her first name. if it was not my aunt, say, my boyfriend's, I could say: "We should've complimented your aunt on her coffee set." So 1) we need a possessive before the word aunt to know whose aunt it is, and 2) we compliment people *on* something in English. Otherwise, great -I want to see this coffee set!

It would be more natural to say, "we should have given our aunt compliments about her new coffee sets".