They do tend to be pretty American, I really don't notice a lot of Europeans wearing them around the city. That said, we just bought a bunch for the Deutsch Wandertag last month, which is a quintessentially German event, and my husband picked up several at second hand shops in Lithuania when we lived there. I think the difference is, in Europe, men wear baseball caps when they're working or doing sports, but not just around the town. So if you were hiking, you'd probably be in good company, but if you were wandering the streets of Cologne, you might be more readily identifiable as a foreigner. I guess it depends, too, on the cap you bring. If it blares Yankees, Cubs or some such, yeah, everyone will immediately know you're American. My husband usually opts for his Wandertag or Vilnius Smiling caps when he really wants to wear a hat in the local community, or bless him, the Whitecaps hat I bought him last time I was in Vancouver.
You don't really notice a lot of Europeans wearing hats most of the time in summer, though older people still do (especially the traditional German ones) or if they're working in the fields or whatnot (we live in the country, we see a lot of people working in the fields).
My husband has a Tilley hat he really likes, though we have heard the comment that only old people wear Tilleys. I don't know, I've had mine since I was 22, but we are admittedly older than most backpackers these days!
Basically, though, you're likely to stand out as a foreigner anyway, so if you want to bring along a cap, do so - some days you won't want to wear it, but other days you won't care about being identified as a traveler. If you can get a cap with a FIFA or World Cup logo, that might be a good compromise!
:tumbleweed: