It's true that there's a suffix spelled
-y which forms adjectives. This suffix looks the same, but it's actually different.
This is a hypocoristic or diminutive suffix
-y which can be used on nouns (including proper nouns). It does not change word class to adjective. In this case, fishy is a noun, and so is dishy.
So what's this suffix do, then? Basically, it makes stuff sound cuter.
This suffix turns dog into doggy and pig into piggy. Sometimes the sound of the base word changes slightly, so cat becomes kitty. For example, if you decided to have a conversation with a piglet, you might say something like this:
Who's a cute wittle piggy!? You are! That's right, yes you are!
(Note the change from little to wittle. This, too, makes stuff sound cuter.)
This suffix is also spelled
-ie and is often used to form pet names. (That's what hypocoristic means.) And if you attach it to a multi-syllable word, you usually get rid of every syllable except the first. For example, you might call someone named Christine Chrissie, first shortening the name to Chris and then adding the suffix.
None of this is acceptable in formal Standard English, but it's acceptable informally.