Better to be safe than sorry....
I would certainly suggest carrying a parent/guardian notarized letter as described above, and pinniped's student ID seems a valuable adjunct (potentially saving a little money, here and there - and Europeans are into "student status", with grey-bearded, middle aged "students" in evidence). Depending upon your "look" (dress, demeanor, companions, etc.), many US hotels and some of the more upscale European hotels may be age-sensitive. Europeans do tend to accord "adult" status for many activities at an earlier age than do US jurisdictions. Having grown up and traveled in a world pre-debit & credit cards, perhaps I'm less secure than many, and always travel with 2 working CCs and 2 DCs.
The old rule of thumb: If you were tall enough to put your coins up on the counter/bar, most places in Europe would serve/accommodate you.....