Solder itself can't expire, but the flux core...maybe?
Honestly, that's probably going to be the most significant difference between solder no-name and name brand solders. The no name stuff often won't even call out the type of flux core... There are a TON of different fluxes out there...water soluble, rosin based...some other stuff...
Now...the purpose of flux is to clean the surfaces that solder is being applied to; with that being the case, I imagine there *could* be some amount of molecular transformation that occurs over time that could cause the flux core in a spool of solder to become a *less* effective cleaning agent.
This would primarily show itself as a solder that resisted wetting to a surface...although that symptom could pop up for a MULTITUDE of reasons. I imagine that a reasonable way to determine the effectiveness of the rosin core would be to use a flux pen on one joint and none on another, soldering both, and comparing the results.
I'm sure the elemental proportions could be a little bit fudged on some of the off brand stuff, but both lead and tin are relatively cheap; I doubt an off brand would fudge those numbers on purpose unless they were making an alloy with pricier raw materials, EG; AU, AG, CU. Perhaps an off brand would be more likely to include larger proportions of impurities, IDK.
Though TBH I don't much care for silver solders on circuit boards. Guitars, offboard stuff, sure. Copper Piping: HELL to the YEAH (I wouldn't use anything else, but at that point I'm specifically looking for a non-eutectic solder)
...sometimes I just type stuff out and forget my point. Solder expiring: maybe? Probably would have done well enough.