Hi there,

Just catching up to your post and have only a couple ideas for you to ponder.

I have never worked on the Volvo rear ends but I have some experience in manufacturing gear boxes that were a little bit bigger.
Like a few in particular that you could park a VW bug down inside of one if it had no gears in there. We use a lot of herringbone reduction gears since these were that type of gear boxes.
We had to have good fit up or "strike" along the length of the tooth of the mating gears.
We observed this by using layout dye, Hi-Spot "Prussian blue" paste or even some lapping compound to create an artificial wear "scuff" pattern to make final adjustments. These were high horsepower or high torque output industrial boxes. Use on aluminum rolling mills and the strength of the drive depends on full face width contact.

The term you are describing is called "backlash." In a properly setup gear train that is kept to a minimum but there has to be some to allow for expansion of various components.
I don't think a differential "Needs" all that much so I will say, there is too much "Tag play" between engagement from one direction to the other. More likely Something may have shifted inside the pumpkin or some highly excessive wear has happen.

A pry bar and some dial indicators inside the pumpkin may be next.
Putting in fresh thick oil might work, if you think there is water in there now for an oil!

Food for thought, if there is wear, the pinion gear turns around on its teeth far more times than the ring gears.
There are, on some rear ends, a "crush" block or sleeve that pulls the pinion forward up tight. You might check into this type differential to see what it uses. Maybe it was not crushed enough and now there is play.
Some need around 150 foot pounds of minimum torque on the nut and sleeve.
The seals should have been weeping some by now at this age. Someone may have been in there once before and it's not set up tight enough forward to engage the angles for a complete strike.
With two angles meeting, the lateral movement is not very much to make a lot of backlash.

Just trying to help as I have limited experience with automotive drive trains.

Phil