The only mistake is the mixture of formal and informal register.

The request 'May I speak to Alice?' and the response 'Speaking' are formal English, while the greeting 'What's up?' is very informal. You would never use these phrases in the same context.

Here's an example of a formal phone conversation, for example, in a business context:

Hello. May I speak to Alice?

Speaking. Who's calling, please?

This is Dan Wilson from CLA Plastics.

Hello, Dan. How can I help you?

And here's an example of an informal conversation between friends:

Hi. Is Alice there?

This is Alice! Who's that?

It's Dan.

Hey, Dan. What's up?

I hope that helps.

What do you mean?

Why on earth did you ask me this question, if you thought that you already knew the answer?

You might want to look here:

The first paragraph explains why "This is she" is wrong.

And yes, it is an American site.

<em>May I say "This is she"?</em>

No, I wouldn't say that. It doesn't sound natural at all.

There are some people who make responses of this type, especially when they are trying too hard to sound formal and correct. I would not recommend copying them.

Please, stop making a big deal about nothing