There could have been people saying the two games are similar about the time these games were originally released here. As RPGs weren't as plentiful as they are today, Dragon Quest (then-Warrior) came out about a year or two before Final Fantasy and since people weren't too used to RPGs, I guess they generalized the supposed similarities.

Random battles, a top-down angle view, buying stuff, towns, character building, talking to a king, rescuing a princess, crossing a bridge into more dangerous territory, etc. all the simple stuff that people experienced after playing Dragon Quest back in the late 80's probably gave them a different look at Final Fantasy when it came out. Both of them had cult followings and [I]Nintendo Power[I] magazine was heavily promoting these games. Its surprising this subject has come up because I bought both games in '94 (DW-used, FF-brand new!) and didn't really see much similarity.

Its only a part theory as to why some may see the two games as similar, but obviously there are plenty of differences that keep them from being too similar.