Meaning – ‘What Does She Look Like?’ vs. ‘How Does She Look Like?’

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:

• I like chocolate - here, like is a verb. It answers the question "what to do?" and its meaning is "to enjoy", "to sympathize" or "to prefer".
• I am a programmer, like my brother - here like is adjective. It answers the question, "how?" and its meaning is "similar to" or "close to".
• There are lots of birds like ducks and gulls - here like is an adverb. Its meaning is "to give an example".

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.

The difference between the two phrases is that both can be used in the literal sense (to look in the specified direction), "look to" and "look up to" also have a figurative meaning.

The literal meaning:

Look at that, Henry's taking his first steps!

Look to your left and your right before crossing the street!

Look up at the sky, it's Superman!

Look up to your right, there's a green bird at the very top of that tree!

In all of these sentences, "look [x]" is being used to mean "look in the direction specified." This is the literal meaning of the phrases.

However "look to" and "look up to" also have figurative meanings. You can say you "look to" someone to find answers or advice, for example:

In her first few weeks on the job, Monica often looked to her boss for guidance.

This doesn't mean that she literally looked at him, but that she went to him for advice when necessary.

"Look up to" also has a figurative meaning. To look up to someone is to see them as a role model, or to view their behavior as a higher standard which you would like to achieve.

Tommy had always looked up to his baseball coach. He was tough but fair, and taught the team a lot. Tommy had wanted to be just like his coach when he was a kid.