There is a coincidental correlation here that can be very misleading about the reasons, and makes your observation partially wrong.
There are two uses of "milyen", similar to regular adjectives: a predicative use (when it belongs to the copula verb "van") and an attributive use (when it belongs to a noun).
On one hand, English uses completely different structures to express these two uses of "milyen". For the predicative use, you have "What is it like" and for the attributive use, you have "What kind of X".
On the other hand, as some of you may know, in Hungarian, predicative adjectives can and should be inflected so they can be pluralized if needed. Attributive adjectives never get inflected.
And this is why it could seem like plural is for one meaning and singular is for the other. "Milyen az a fa?" belongs to the "What (...) like" type, yet it's singular since the subject itself is singular.
Is there a way to ask "what kinds of trees" vs. "what kind of trees"? In English it seems like "kind" and "tree" should both be plural or both be singular, but it is not consistent.
Should we translate "milyen fák" as "what kinds of trees" and "milyen fa" as "what kind of tree"? Now I"m not sure I know English any more...
Because of Hungarian grammar, this question simply cannot come up. Think of it the following way. "Milyen" can be answered with a usual adjective like "szép", right? Now, in Hungarian, "milyen fa" and "milyen fák" are pretty similar to "szép fa" and "szép fák" grammatically. In this case, even in English, you would probably just say "beautiful/nice tree" and "beautiful/nice trees" and there is no "kind" to worry about.