Definition of volume of a gas and Amagat's law How would you define the volume of an (ideal) gas? 
I would say it is the volume of the container holding the gas (whether it is a fixed container or a flexible one like a balloon).
However, assuming this is the true definition of the volume of a gas, is it true that Amagat's law of partial volumes is merely a mathematical statement and has no physical significance? What does the volume of one gas in a mixture mean? 
Each gas still takes up the whole container. 
(If you would give a physical interpretation, wouldn't it be the following one: The volume of one gas in a mixture having a certain temperature, pressure and number of moles is the volume of a container hosting only one gas having that temperature, pressure and number of moles.)
https://en.wikipedia.org/wiki/Amagat%27s_law
 A: So, I’m not sure I know the “right” answer, but I’ll take a crack at it.
So, first, I think it’s a good point that we don’t measure partial volumes. The volume of a gas mixture is what it is. 
But, we also need to recognize that (I don’t think) we measure partial pressures directly either. You can certainly construct something like a pressure gauge that only selectively measures pressure from one constituent of a gas mixture, but ultimately what your pressure gauge will do is measure a gas density and relate it back to pressure by some algorithm, probably ideal gas laws with appropriate corrections for the constituent of interest.
So, what do we mean when we say partial pressure? I think we mean the pressure we would observe if all the other constituents are removed and all the other thermodynamic properties are held constant.
The same is true of this idea of partial volume. It is the volume the gas would take up if all the other constituents are removed and all the other thermodynamic variables are held constant.
The whole concept of partial pressure and volume is only possible because the effects of the different constituents are independent of each other. That is, to first approximation, gas A doesn’t know about gas B and vice versa. If the gasses are strongly interacting, either because there’s some chemical reaction going on or our gas densities are really high, this whole concept of partial anything breaks down and we have to consider a more nuanced way of thinking about the effects of the gasses on their enclosure. 
But for simple gasses at reasonable densities,  you can consider the effects of the gasses as more or less independent of each other and talk about their “partial” contributions to the total observed pressure and volume. 
A: To confirm that the answer by user218912 is correct, here's an excerpt from [1]:

The partial volume of one of the components of mixture of gases is the volume which that constituent gas would have occupied if it had the same pressure and the same temperature as that of the mixture.

This is analogous to the definition of partial pressure [2], with the roles of pressure and volume interchanged.

References:
[1] Section 23-7 in Bhatnagar, A Complete Course in ISC Physics
[2] https://en.wikipedia.org/wiki/Partial_pressure
