Is the amount of play normal, or is it excessive? It's a 2002 Stilo 1.6:

might be excessive did you check the rubber if its cracked or cut? thats how i usually do it

that is excessive and will only ruin your flex pipe.
The job can be a bit of a pig, technically easy but that bolt in the middle is a real sob to remove.
That mount was the cause a 3 broken flex pipes on our old car before i realised the problem.
get it changed.

That's excessive on any car.

Amount of play should be measured in mm's, not inches!

Thing is, though, this mount was changed only a year and two months ago. Back then I immediately noticed a massive improvement compared with the old, worn-out mount. But then a few months later I had the gearbox itself changed, and after that the vibrations seemed to reappear.

So I'm wondering, is the mount likely to be shot after just a year, or may the mount be OK, but not be fitted properly? I tried to tighten the bolt myself using a wimpy little spanner, but the bolt wouldn't budge.

looks like a dodgy pattern part tbh, i had wishbones fail only in months because the rubber wasnt man enough.
go to main dealers and get a proper one (or at least an oem made one), this way you know you wont have to revisit the job for some years.
ALSO get a replacement bolt as i found they are very very tight, plus its good practice to replace these sorts of bolt that are under high loads.

Tightening the bolt wont make any difference the rubber is naff