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As per Wikipedia,
Species of identical particles include, but are not limited to, elementary particles (such as electrons), composite subatomic particles (such as atomic nuclei), as well as atoms and molecules.
Therefore, two hydrogen atoms should be identical. But are they if they are in different states? If No!, then does that mean here that the energy of hydrogen atom is its intrinsic property like mass, spin to electron?
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.
This is similar to a question like
Are a spin-up electron and a spin-down electron identical particles?
What differs is states of the particles: spin projections in this example, orbitals in the OP. The particles themselves are still the same species, thus indistinguishable.
