I interpret it as that being the "kind" of fondness this person has. It might make more sense if you put quotes around it or add という.
「お嫁さんにしたい」の好き
お嫁さんにしたいという好き
お嫁さんにしたい is being used as a phrase to describe what kind of 好き it is but not in a way that follows the normal rules of grammar, or to put it more precisely, not in a way that follows how you might expect the phrase "お嫁さんにしたい" to be used. That is to say, it's not "I like you as a friend" but rather "I like you like 'I want you to be my wife.'" Similarly, 好き is not being used in its normal sense of ~が好き but rather as a representation of the word itself. So for example, in English you can say "cats are cute," but you can also say "'cats' is a cute word" and still be correct even though you're saying "cats is."
Basically these words and phrases are being nominalized and used as you would expect a noun be used.