¿Cómo te llamas? translates more literally to How are you called (in general, maybe by others)? than to How do you call yourself?, since it's a reference to a pronominal verb (a verb that needs to be conjugated with a pronoun, and in this specific case it was formerly called pseudo-reflexive), contrary to the call in your example, which is simply transitive. But in practice it just means What's your name?.
llamarse reflexive verb :
to be called, to be named
<¿cómo te llamas? : what's your name?>
You have a very similar verb, apellidarse, which has the same construction:
apellidarse reflexive verb :
to have for a last name
<¿cómo se apellida? : what is your last name?>
This case is more obvious, since there is no confusion between the two different flavors (transitive and pseudo-reflexive/pronominal) of the same verb. Despite the existence of the transitive version of apellidar, it's not as common as the transitive version of llamar.
llamar (a alguien, transitive) vs llamarse (preudo-reflexive/pronominal).
apellidar (a alguien, transitive) vs apellidarse (pseudo-reflexive/pronominal).
Now, regarding the last part of your question Cómo te llamas is more likely to be used in everyday speech and Cuál es tu nombre, since it's a little bit more formal, maybe also a little bit less used.