Some people do, some people don't. What I wouldn't do is mix "high standards" with "looking for compatibility".
It's difficult to define what a high standard actually is, because our needs and desires are all unique to us as an individual. What someone may consider to be a high standard may be another's preference, and vice versa.
For example, because of personal experience I can identify characteristics I would consider a must in a partner, as well as those which I would prefer and those I would stay away from. Thus, my standards are "higher" because I now have a better understanding of what I am looking for, and therefore I would like to think if I were to meet such a person the relationship would be stronger and healthier as a result of being more compatible. I do take physical features into account, but in that regard I have always been pretty diverse and so long as I found them sexually appeal (IE. Attractive) I would not rule them out. My preference is red heads, but I wouldn't turn someone down for being a brunette.
Of course, there are people that insist on someone being X height, having Y color hair, interesting in Z music. Personally I don't understand why people restrict themselves so much, but sadly this is modern dating culture and a reflection on how, particularly in younger people, the "grass is always greener".
I'll say this, and I don't say it out of bitterness, but I have noticed throughout the years that it is women who have far more requirements/standards than men. It could be modern culture, it could be biological, but I've witnessed first-hand men being rejected for being two inches from a woman's preferred height or not earning enough money. On the other hand, again from what I've witnessed, men are generally more flexible. Of course, this isn't an accurate representation of dating but just what I've come to see for myself.