Why is the voltage across my nichrome wire so low?
I want the wire to reach 100°C and datasheet (RD 100 / 0,6) says that needs 2.21 A, ...

Correct. And since the resistance is 1.73 Ω/m you need 3.8 V/m of wire.

... according to ohm law that's 24 W ...

That's Joule's law, P = VI.

... and with 3 V across it that needs 8A (I = P/V).

No, R is fixed so you can't just set the voltage to any value like that. At 3 V you get 1/4 the voltage you get at 12 V and that gives 1/4 the current resulting in 1/16 the power.

Now my question is how I can increase the V across wire so that less current would be required?

The way to do that is to draw out the wire so that it has 1/4 the cross-sectional area (1/2 the diameter). The resistance will then be 6.9 Ω/m.

Why the V is so much less than the supply?

Because you have overloaded it.

I obviously need a supply with higher amp but with an actual 12 V across wire I would only need 2 A and supplies with lower amp limit tend to be cheaper.

There cheaper because they are not as powerful, of course.