My 3speed automatic transmission started to leak and took propeller shaft to see what's going on. I found out that rear seal was in pieces, but the didn't harm the rear part of gearbox(new seal is going to be tight). I was trying to figure out what caused that and noticed that the axle which is inside tha gearbox is very loose. Should it be like this or should it be tighter? Is there a bearing which holds the axle steady?

There may be a little play in the shaft but certainly not a lot!

I suggest if the shaft has a barely perceptible movement, that would be tolerable.

If the output shaft has excessive wobble it will allow the drive shaft slip joint to rotate in an uncontrolled eccentric pattern which will chew out the rear seal again.

If you are genuinely concerned it may have too much movement, it may be wise to have a transmission expert check it out.