Suppose the separated volumes of identical gas are a low-entropy state and the mixed volume is high-entropy. Imagine the reverse process from mixing. You have a single tank full of helium. You insert a partition, so now you have two half-tanks of helium. This can be done reversibly, but it takes you from the high-entropy to the low-entropy state. The entropy decreased without any increase of entropy elsewhere in the universe, so you have violated the second law.
The same argument doesn't work for different species. If you have mixed helium and argon and insert a partition, you'll simply have two volumes of mixed helium and argon, so you haven't gone back to the low-entropy state you started in.
Another way to say this is that if mixing identical gases gives an entropy increase of $\Delta S$, then you ought to be able to get work out of the process. Specifically, you should be able to convert $T\Delta S$ Joules of heat from the environment into an equal amount of work. But what physical mechanism would let you do that?
If the gases are different, you can use a semi-permeable membrane. Set up the gases with helium on the left and argon on the right. Make the membrane permeable only to helium. Make the membrane free to translate. It will start moving to the left as helium moves through it to the right to mix with the argon. You can apply an external force to the membrane and it will still move as long as the force is not too great. In this way you extract work. But if you have helium on both sides, the setup fails; the membrane won't know which way to go.
If the molecules are all distinguishable, then these arguments fail. That is, if you can say "molecules 1, 3, 5, 7 etc are all on the left and molecules 2, 4, 6, etc are all on the right", then allowing them to mix does cause an entropy increase because you lose information about the molecules' position. You can still insert a partition into the gas, but you won't be able to do it in such a way that the molecules are separated the way they were before. They'll just be randomly mixed. So ultimately, whether the entropy increases or not is about whether you're losing information. Having the molecules be different species simply gives us a way to tell them apart, and this distinguishability is what makes the mixing irreversible.
Both the gases, though are same, when expand, make their entropy increase, isn't it?
Yes, if you had two individual volumes of helium and you let both of them double in volume, that would be an entropy increase. But that's not what's happening. When gas expands, entropy increases because you know less about the positions of the molecules. Imagine two separated liters of helium. Choose a molecule. How accurately do you know its position? It could be anywhere in the left liter or anywhere in the right liter. You have an uncertainty of two liters. After the mixing, you still have an uncertainty of two liters. Nothing has changed.
With the two different species, argon on the right and helium on the left, entropy increases because each helium molecule's position uncertainty goes from one liter to two, so it's a different scenario.