What is procedure of calculation for the volume of certain objects, one of 10 kg and another of 100 kg, with an approximation of 10%?

You need to know the density of what you are measuring. 1 litre of water = 1kg. 1 litre of oil is less than 1kg. 1 litre of mercury = more than 1kg. Volume measures are NOT weight measures. Imagine 1 litre of rice vs 1 litre of sugar, or packed brown sugar. They are all going to be different weights.

I should have mentioned it. In this case, they’re both feathers.

I do not see any way to calculate the volume of feathers, because they are really compressible.
Take a pillow, for example. When you buy one, chances are, it will be vacuum packed, to minimise storage space for the manufacturer and distributor.
When you unpack it, it will inflate.
Both states will have the same mass (minus the mass of the air in between it), but vastly different volumes.

Without knowing it’s about feathers, the procedure I would use would be to point out to your dad that kg is a unit of weight, or mass, and that volume is a measurement of space, which are two different concepts, so the question is meaningless and should begin with understanding the terms it uses. “with an approximation of 10%” is also meaningless, and would be the next term to learn about.

Adding that it’s feathers helps, but as @ragingloli pointed out, feathers do not have a standard density, even for the same type of feathers, so there is no way to get within 10% accuracy.

If the material were something with a much more predictable density, such as water at standard temperature and pressure, then the volume tends to be a simple matter of dividing the mass by the density. Since feathers vary widely in possible density, it’s not possible to be accurate – you’d need some estimated figure for density, which you could get by weighing feathers stuffed into a box of a known volume, then dividing (total weight – box weigh) by box volume to get the density of those feathers in that condition.

I’m with @Zaku on this one. Mass and volume are different parameters. They are only tied together when you add density of a substance and a given form (sphere, cube, pyramid, etc). 10 kg sphere of lead will be a much smaller volume than a 10 kg sphere of carbon.

You can look up the density of some kinds of feathers, then assume those are the feathers he is talking about. Then you can calculate volume knowing 10 kg. Multiply that volume by 10x for 100 kg. Accuracy of 10% is not achievable because you don’t know exactly what feathers he is assuming and because feathers are compressible, so 100 kg of feathers might compress to be less than 10x of 10 kg.

‘Feathers’ is too vague an object to compare. You need to know the specific density of what you are measuring if you expect to make a direct comparison between volume and weight. Look up the term ‘Specific Density’ and you’ll get closer to understanding what your father is trying (but badly) to explain to you.

We will need to know the shape and how much they are compressed…