Can I replace the SSD in my late 2013 MacBook Pro A1502 with a second-hand model from a different year?

The others have given great advice.

I would just like to say that the SSD will physically fit into the SSD slot on any of the Late-2013 to 2017 Macs (without USB-C ports), but macOS may not work correctly or even boot when you do that. I have first hand experience with this happening. The same SSD worked fine when using Linux, so the problem was with macOS itself not liking having an unexpected SSD model installed in certain Macs.

Unfortunately there is no easy way to identify which PCIe blade style Apple OEM SSDs will work in a particular Mac. The solution proposed by the others is the easiest way forward if you don't have a good backup. If you don't have a good backup, then I highly recommend you start now before you lose some important data.

This is really tricky.

It's been a while since Apple transitioned to somewhat (or sometimes entirely proprietary interfaces for storage. Long gone are the days of 3.5" (or even 2.5") SATA drives that you could just pull with impunity.

Specifically, different form factors, different interfaces and, in some cases, lack of support from Apple make what you're asking kind of tricky.

Plus, you're proposing getting a newer machine and relying on a 10+ year old drive to run it?

I'd probably take a slightly different approach. Get an OWC Envoy Pro from: https://eshop.macsales.com/item/OWC/MAU3ENP1AW/

it's an external enclosure that you can put the drive in to, then attach it via USB to any other system. You can then either retrieve the data, or continue to use it as an external volume. At least this way you're free to get any other system you like, without needing it to be hardware-compatible with your old drive.

a similar solution with slightly better portability:

envoy express. same vendor.