When we ask whether two particles are identical, we ask whether they have intrinsic properties that allow us to distinguish them. We do not ask whether they can occupy different states.
The example that can be typically found in textbooks uses the position of the particles as their state. We learn that, for indistinguishable particles, we can only say "one is at $x_1$ and one at $x_2$" but not "particle A is at $x_1$ and particle B at $x_2$". It works the same for the electronic state of hydrogen atoms: we may say that one atom is in the state 1s and one in the state 2s but we may not label them.
Confusion might arise because hydrogen atoms have both the electronic state and a position and we are indeed able to say "the atom at $x_1$ is in 1s and the atom at $x_2$ is in 2s". However, of the two atoms in the states $|x_1, 1s\rangle$ and $|x_2, 2s\rangle$ we can not label them and say which is which.