If someone, who is not grateful, is "ungrateful", why is he called an "ingrate", and not "ungrate"?

That’s English for you! (And also French and…...)

Because he expresses ingratitude instead of ungratitude.

Good question.
Like @janbb‘s answer. Modern English has plenty of origins, and can be pretty complicated.

English is annoying and illogical a lot of the time. I once saw a statistic that 20% of English is irregular, or exceptions, or something like that.

Ingrate is a noun “He is an ingrate.”

Ungrateful is an adjective “He is ungrateful.”

Because they entered the language separately. “Ungrateful” is “un-” + “grateful”, “ingrate” is borrowed directly from the Latin word “ingratus” meaning “unpleasant” or “disagreeable”.

^ Excellent.

Plus, and “ungrate” is a wedge of cheese not suitable for grating.

^ I thought it was just when you compressed the bits back together…

^ Ah yes, reconstruction.