When 'as in taste', it is almost same as in English.
There are indeed expressions, but some same, some very different. あまいことば = sweet / pleasing words. こどもにあまい = be too kind to the child(ren). 採点(さいてん)があまい = scoring (on a test) is tolerating some subtle mistakes / fuzzy answers. あまいかんがえ = over-optimistic thinking. わきがあまい (lit. 'the armpit is sweet') = too loose so that the opponent takes advantages / is benefited.
Each word is pronounced differently, depending on the pitch of a character it could change the word. For example the word hashi, if one pronounces Ha with a lower pitch and Shi with a higher pitch it is the word "chopsticks", and if the pitches were vise versa it would translate to "brigde" or "edge"
I thought it denoted someone's sweet nature or appearance since I knew this word from the superhero Amai Mask from One Punch Man... now I know that his name doesn't mean something as deep as "he's an evil man wearing the mask of a sweet caring superhero", but that it's talking about a mask that literally tastes sweet? XD
So a couple weeks ago i googled some of the words on here like i googled ohayo and hai and stuff in their romanized forms, and i googled amai bc a friend whos ahead of me on duolingo had told me about it, and ive never forgotten, that its like some sort of mexican association or something, and it helps a lot bc i think mexico, spicy, halfway across the globe or whatever, Japan, sweet. So thats how i remember this one
It can also be used metaphorically for a naive person, saying someone is sweet is saying life has not hardened them or made them bitter yet xD.
It's used this way a lot in anime because, you know, generally everyone is pretty honest and plays with their cards facing up in anime right? /sarcasm