Can I keep the OS as-is, on swapping SSD on compatible Macbooks? I own two Macbooks, an Air mid-2013 (256gb) and a Pro Early-2015 (128gb). As far as I know their ssd drives are the same model.

The question I would like to ask is: 
Can I take the SSD drive from the Air mid-2013 and install it inside the Pro early-2015 and use the OS, with all the data, as was installed on the original Mac? Or would I need to reinstall the OS on the newer Mac?
Yes, you can replace it without reinstalling the OS and keep the data. 
The only issue you can face is a long time first boot.

But I recommend you to backup both disks to external drive like a Time Capsule or external hard drive before replacing.