"My older brother likes purple the most."
紫色的 is an attributive since it has the marker "的", which implies that what your brother likes is not purple (the noun), but purple (the adjective) plus a omitted noun. So your sentence actually means:
My older brother likes the purple one most.
That sentence should be changed to: "My older brother likes purple color the most"
In a conversation, Yes. In written Chinese, I have not seen someone write the 的 afterwards. It is acceptable in conversational Chinese because these 的、了 are sometimes just for expression of feelings or casuality, without a concrete meaning.
Note that you can also say 我的哥哥最喜欢紫色的 in conversation, in case you are trying to move the 的 around.