Yes. Though not as fast as a tire out in the sun. Tires in the sun will suffer UV damage as a primary form of degradation (obviously mechanical were too if in use). Spairs don’t see the sun so not a problem, but they do oxidize. Ozone will attack the rubber in the tire and cause it to go brittle. that is what causes all the tiny cracking you see on old tires. Generally it is recommended to replace any tire over ten years regardless of use. Interestingly this applies to brand new tires that have been sitting in a warehouse. It is worth learning how to read a tire date code so you can check the tire age. This helps with buying used cars, making sure a tire company didn’t sell you NOS tires, and just general safety.
For you I would just swap it out next time you buy tires. If it is a full sized spare then just save the best of the tires you are replacing as the spare. Save you buying a whole new tire that you will probably never use.