Monterey and Lightroom 5

In another thread @Jim Wilde pointed out that he has tested old installers and found that the Lightroom 5 installer is not 32 bits but 64 bits, so it will work on an M1 Mac with the help of Rosetta 2. Apparently the 32 bit problem is only in the licensing part of the Lightroom 6 installation (Lightroom 5 did not have online activation yet). Having said that, I would advice against upgrading your system if Lightroom 5 is so important to you. It works now, so why take the chance to create a problem? But maybe you should not connect your computer to the internet in that case, because of all the security risks of an old OS...

Better still, forget about Lightroom 5 and bite the bullet. Get Lightroom Classic and enjoy all the improvements that Adobe added over the years.

You've posted in the Lightroom Classic forum so we all assume you're referring to the old Lightroom 5 from 2013.

Not the current Lightroom (Cloud) which is currently version 5.x.

LR5 should run as a 64-bit application under Monterey (within reason) but installing it is the issue; you must boot into an OS that can support the 32-bit installer (the last being Mojave). That's a big problem for M1's because Monterey, Big Sur, Catalina and M1 Macs do not run 32-bit and can't boot into Mojave. 
On my 2019 Macbook Pro (Intel) I can run LR5 which was installed long before I updated that OS, using an older OS that supports 32-bit installers. That said, the Map module will not work and there are other issues, you really should consider a modern version of LR instead. 

Not updating your operating system exposes you to risks. So update your operating system and upgrade Lightroom 5 to the current Lightroom  Classic.