Do we have to treat ideal gas as identical particle?

In Quantum Mechanics 2nd edition by Griffiths, he says we can get Maxwell-Boltzmann distribution as we deal with distinguishable particles. And I think ideal gas follows MB distribution; therefore, I think ideal gas is distinguishable until now.

However when I start learning thermal physics, the textbook tells me if a system has N gas, and for each gas it has Z (partition function) individually.
Then for the N-gas system, the partition function is $Z^n/n!$.
Because Gibbs treats it as identical.

So, the question is …
the ideal gas should be treat as identical or not??P

(Actually I have a thought experiment for this..p)

If we have 4 ideal gas particle in a room with volume 2V, and I separate the space as V+V virtually.
Evidently, we know each volume V could have 2 particle in average when it reach equilibrium.
And my reason is it has only 1 configuration when all particles are in the same V, but it has $$6=frac{4!}{2!2!}$$
When the gas are distributed equally in the 2V. it has the highest multiplicity, which we can infer it has the highest entropy.

However as I think like this, I’ve already treated it as distinguishable.