Do magnesium and tin have the same volume?
No, that is not how it works.

Atomic radius means the radius of an atom. A single atom. It does not in and by itself mean that a solid chunk of a material has a particular density. A solid chunk of material has many atoms, and they are arranged in certain ways and sometimes more than one way of this arrangement is possible.

Tin and Magnesium are very different. Magnesium is arranged in a hexagonal close packed structure, basically just repeating units of skewed (rombohedric) prisms. This can be hard to imagine, but imagine just stacking perfect spheres so they rest on each other and likely you will end up at a close packing arrangement. (there are only two*, try to find them. Hint: one has three layers before it repeats the bottom, the other has only two layers then it repeats)

*: There are actually 4, but you'll need a lot of layers and a lot of spheres to find those.

Tin has two different ones, so already here your question has to be no'ed without further investigation, but lets anyway. The first one, grey tin (${\alpha}$) has a diamond like structure. I am finding it hard to describe them in a meaningful way, but they have examples over at the excellent Wikipedia. The other one, tin (${\beta}$) which is white has a body centered tetragonal - which is basically stack 4 - put 1 in the middle and then stack 4 on top of that again (yes, the spheres will roll off, but there are so many of them per layer that hardly any of them do.) The white one is amusing, if you bend it it screams like you are hurting it. Sounds absolutely disgusting, like nails on a chalkboard. This is related to it's structure.

Summary:
When you put a whole mole (ha... ha...) of atoms together, they organize in structures, and it is the number of atoms per volume that you have to multiply with the atomic weight to get the density of the chunk. Fortunately for most kinds of structures, you don't have to do this, someone already has done the work, by actually measuring the densities of crystals and made tables. Just look it up.