Context is everything...
I think maybe we are knee-jerking a bit too much on this one.. I don't see how it would be racist to do this with a number of races or nationalities (yellow hair and viking helmets for everyone?) but I believe the context here is that americans (probably) are complaining about an App which unfortunately stereotypes a certain kind of abusive racism prevalent in america during the last century, where they portrayed asians (particularly chinese) in movies as blundering idiots with straw hats, squinty eyes, L>R on the fly translation, funny clothes and somehow always wearing traditional chinese clothing and fortune cookies everywhere. To avoid stereotyping some of them also worked in laundry shops.
This negative stereotyping was reinforced by a movie industry which couldn't pick on blacks (too much), because, you know, slavery, but asians were fair game, as they had only really came over to the white man's land to steal white jobs on the railroad anyway.
It is most likely against this context of the over the top asian stereotypes that the complaints came in; not that the app gives you slightly slanty eyes, dark hair, etc, but that it contains these character traits (straw hats, fu-man-chu mustaches, etc.. ) that for so long helped to ridicule asian americans and lead to, yes, racism. It is perhaps not that well known outside the US.
For easy to understand context, imagine a make me black app (not racist in itself.. at least I wouldn't find it racist.. others might), but which had the added bonus of being able to put you in cotton fields, slave chains or similar, which would then, because of context, be racist.