SPCR

I'd like to use a piece of tempered glass from a section of deck railing as an office chair mat.

Is this possible? Is all tempered glass (e.g. from furniture, shower stalls, deck railing, chair mats, etc) more or less the same in hardness and durability?

I'd be surprised if all tempered glass is the same, but I see no reason why a panel intended for a railing shouldn't work as a floor cover, provided the floor underneath isn't too soft.
A railing panel should be made to survive mechanical stress such as hard winds as well as some kicking and beating.

You could also consider what's the worst to happen if it doesn't work: The glass breaks up into smaller pieces and splinters on the first use.
Is it okay for you to wear shoes and be ready to sweep splinters off the floor at the first usage?
If it doesn't break quickly you'll get notices by cracks, so after the first use there should be no significant problems.

Depends on the quality of the glass and how flat the floor is. If the leg of your work chair falls into the place under which there is a recess, the glass may crack. Do not forget that the pressure on the glass will be very limited in area, which will increase the pressure. This is similar to walking on the ice just in boots or putting on snowmobiles. In snowmobiles, the chance of falling is lower. The same is with glass. It can withstand pressure over a large area, but not the same weight when applied to a point.