This is correct:
Some people of the same nationality, the same age, the same sex, and the same height are in the room.
This is also correct:
Some people of the same nationality, age, sex, and height are in the room.
Even this is correct:
Some people of the same nationality, same age, same sex, and same height are in the room.
The only thing that is not correct would be this:
Some people of the same nationality, the age, the sex, and the height are in the room.
Whichever of the first three you should choose mainly depends on style and what you want to emphasize.
The first one is probably the least natural, and its emphasis is on being wordy/verbose.
The second one is focused on being short and to the point, but it risks focusing too little on the word "same". It's probably the best for many casual settings.
The third one probably strikes the best balance in a formal setting. It's not unusually verbose, but it is verbose enough to sound formal. And it also places a fair amount of emphasis on the fact that everything is the "same". It could also be the best choice in casual settings, when the speaker just wants to place a lot of emphasis on the word "same".