Mac won’t stay asleep or shut down properly

QuiGonJinn is on the right track with suggesting an overactive Bluetooth connection, but just beware if you use Bluetooth Settings to "forget" that keyboard, then you will not be able to use that Mac until you connect another keyboard to it. So be sure to have another one ready to go first. You can use any inexpensive generic USB keyboard for that purpose.

If a Mac does not shut down or sleep properly, If your Mac won’t go to or stay in sleep - Apple Support describes the applicable steps to follow. Reset the SMC of your Mac - Apple Support is also justified for that problem.

Have you tried turning off Bluetooth before shutting down the Mac? If it sleeps/shuts down like it’s supposed to, then you’ll know it’s a faulty Bluetooth connection. Either the Mac, or the keyboard may be sending each other too many handshakes. (Keeping each other active and awake) A solution for that issue would be to make the Mac “forget” the keyboard through Bluetooth settings, and then reconnecting it.

You can also try this, if your Mac is Intel based.

Good luck and tell us if it works.