What I mean by classical: For the case of $GL_2$, the answer to my question would be that the automorphic forms are either Maas forms or modular forms. For $GSp(2n)$ these are the Siegel modular forms.

Are there any "classical" objects which appeared in mathematics before this automorphic perspective which align with automorphic forms on $U(n)$ or other unitary groups?

Maybe its the case that, historically, automorphic forms on unitary groups came first: then the question is: are there any other "models" of these kinds of functions?