Why don't I see the "sync" button in my messaging app on my computer in preferences?
__________

That could be explained by the version of the operating system. The sync button appeared after a certain number of the operating system. It shows on 10.15.7 Catalina and later. To find out what version of the OS you are currently running, you can go to the black apple in the top left corner of your screen and choose “about this Mac”. That will then display what version you are on.