Are OS installations on a disk linked to a system? I have a MBP17 with an SSD (Sierra) which I want to switch with my husband's MBP13 for work related reasons.

Would taking out the SSD (OS and files intact) from the MBP17 and placing it in the MBP13 without re-installing anything render any problems? Meaning is the harddrive linked to an installation in any way, leading me to have to re-install Mac on the SSD before I can use the MBP13?

Would it matter that the MBP13 is still on El Capitan and the MBP17 on Sierra?
This shouldn't be a problem. I have done similar things both with an external disk and an internal disk. The first boot may take a very long time, but it is working — leave it to do its thing. Of course, double check your backups before performing something like this.