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.

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.