Is there a name for someone who calls someone else what he himself is?

An idiom for this situation is the pot calling the kettle 'black'.

A fitting term for a person who does this is hypocrite; though the exact meaning of that word is someone who pretends to be what he is not, it applies because the person is pretending to be better than whom he is denouncing, while in reality he is the same.

If you want to look at it from a psychological standpoint, using that terminology (suggested by Henry), you could consider psychological projection (aka projection bias):

Psychological projection or projection bias is a psychological defense mechanism where a person subconsciously denies his or her own attributes, thoughts, and emotions, which are then ascribed to the outside world, usually to other people.

Psychological projection appears to be the most common and unambiguous term; referencing google searches, I find that projection bias may sometimes be used to mean something else.

"Hypocrite"(n) and "hypocritical"(adj) come to mind. There are other expressions describing the situation, although they are not specifically a label/adjective for the person.

The term "projection" may be what you want. (It comes from psychology originally, but gets used in other contexts.).

Three common terms coming from psychoanalysis or psychology are transference (see dictionary.com definition 3), displacement (see dictionary.com definition 7), and projection (see dictionary.com definition 11)

Although psychoanalysts and psychologists may see some fine distinction between them, for the layman they tend to be used interchangeably.

Someone might say "you're projecting" or "you're engaging in transference" or "that's a classic case of transference" or "you've got some obvious displacement issues."