The short answer to this question is "yes".
Obviously, in some sense Japanese people define what is Japanese culture. However, it is also true that, today, all cultures are disappearing into a monolithic modernity, and, from that perspective, there are many Japanese people who perceive "Japanese culture" as something antiquated and slightly foreign that requires study to understand.
For example, based on my experience here in Kyoto, I have no doubt that Japanese people do not see keikogi as simply training clothes. When they do training in non-budo of any kind, they wear western athletic wear indistinguishable from that seen in the US. Keikogi is for Japanese activities.
Japanese people tend to be picky about what's acceptable - for example, people playing golf or tennis both have acceptable "uniforms" as well.
I'd note that the keikogi is a created tradition - not something that belongs to traditional budo, but something that was created by Jigoro Kano. It was created as...training clothes.
So..if Budo can't be practiced without keikogi, what did all those Budo guys before Kano do?
As for me, I probably wouldn't study aikido if it were done in street clothes and without etiquette. Many elements of the world outside the dojo are not to my liking, so I see studying any budo as a sort of... what?... meditation? I can turn off certain parts of my brain in the dojo that I couldn't if we were all wearing shorts and t-shirts with corporate logos or identity slogans.
That's your preference of course, but that doesn't mean that that it's impossible, we do it around here quite often. Once again, if Morihei Ueshiba taught outside without keikogi or etiquette (as he was known to do on many occassions), would it still be Aikido?