Volume of different gas in a container
Your second idea, that they have the same volume, is correct.  Each gas in a mixture of gases stored in the same container has the volume of that container.  Remember that gases are mostly empty space, so it is not a problem that their volumes overlap.

As is implied by your comment about subtracting the vapor pressure of water from the pressure of wet hydrogen, the gases in a mixture can have different partial pressures. Perhaps that is your point of confusion.  

The only case where they would have different volumes would be in a more accurate gas model than ideal gases, something like the Van der Waals model, where gas volume is corrected by a small amount to account for the portion of space that is taken up by the gas particles.

In short, each component of a mixture of gases stored in a given container shares the same volume and temperature and may (or may not) have different partial pressures and number of moles.