What would happen if you were to replace the SSD of your MacBook Air with an SSD directly taken out of another MacBook Air , same generation, same everything. Stock.
It will be as if you swopped Macbooks. It will work perfectly as it was in first Macbook.
You can restore your internal "Macintosh HD" hard drive to an external hard drive using Disk Utility in Recovery Mode, then boot from your external drive on ANY Mac. This is ideal if you want to switch between your home Mac and work Mac with only carrying 1 external hard drive which you can even encrypt before it gets lost or stolen. (Enable FileVault in System Preferences before restoring/cloning)
Assuming that the SSD has an operating system (OS X) installed, then it just means the MacBook Air will boot up from the newly installed SSD and have access to all the apps and files installed on it.
It also means the MacBook Air will have all the user settings of the other owner (eg Admin password, user accounts, desktop background image, system preferences, Apple ID, etc.).
So, assuming there's nothing wrong with the other SSD, the computer will function perfectly, but it does mean you'll need to change the settings to suit you - unless of course it's now going to be used by the owner of the other MacBook Air.