What is the proper use of "do" versus "are" in the phrases:
Do you like me?
Are you like me?
Is there any difference between them?
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The misunderstanding is not about "do" versus "are".
Instead, it is caused by the word "like". It can be either verb or adjective, but also may serve an adverb:
Not very grammatical but a nice example sentence is:
Like my brother, I like fruits, like apples or bananas.
Here, "like" serves three different roles, and the meaning is:
Similarly to my brother, I enjoy fruits, for example, apples or bananas.
Now, back to the question.
"Do you like me?" means an asker wants to know if you enjoy the friendship of him/her.
"Are you like me?" means an asker wants to know if you are similar to him/her by character or whatever.