Photos on MacBook running Ventura vs iMac running Monterey

You can't delete old backups manually. They won't be removed from the Trash Bin.

Also, the Photos library can't be on the same drive that Time Machine uses for backup. TM requires its own, undivided drive.

I strongly recommend you get another drive and put your library on (and ny other files as desired). It must be formatted either APFS or OS X Extended (hournaled) with ownership set to be ignored. Then erase the TM drive, format it APFS and start anew with Time Machine.

If you have a dedicated drive for your Photos library you can have it contain the full sized originals instead of optimized versions (if that's what you've done) and be able to back it up with Time Machine along with your boot drive.

First some information: Photos is primarily a database that keeps track of all your pictures and the details associated with them. It’s like an Excel spreadsheet with the rows being the pictures and the columns being the details, like date, location, faces, albums they’re in, and even all the steps in editing a picture. So Photos never alters an original picture file— when you look at a picture, Photos shows you the original and, on the fly, adds all the alterations listed in the database. This is very cool!

The point of that story is that, when you update the Photos app, like you did going from Monterey to Ventura, Photos had to add new columns to accommodate all the new features. Ventura Photos could not read the old library without the right columns, and Monterey can not read the new database that has extra columns. 

So you can see that there is no way to have a single library that can be read by two different versions of the MacOS. 

There is only one way (that I can think of) to view exactly the same stuff on two versions, and that is to use Photos on iCloud.com. If you turn on iCloud in your Ventura Photos, then all the pictures will be copied to iCloud (possibly taking a week or more.) Another computer with the same Apple ID can connect to iCloud and download those pictures into its own Library, and iCloud with fit them into that system’s database. Of course, iCloud will cost a bit. 

You can, of course, restore the old version for your iMac, but it will be static-- new pictures added in Ventura will not appear there. Using iCloud means you can add, edit, delete, or whatever, pictures on either machine and the same will happen in the other.

You say you have backed up your Library with Time Machine. Is TM on the same hard drive? If so, you may have trouble — unless they exist on separate volumes. Time Machine can mess up a a Photos Library that’re on the same volume, even if TM is deleted. And to avoid damaging the Photos Library an external drive must be formatted in either APFS format or Mac OS Extended (Journaled) format. The volume can not have had Time Machine on it since it was formatted.

Move your Photos library to save space on your Mac - Apple Support

"USB External Physical Volume • Mac OS Extended" is good.

I know that claustrophobic feeling when all your storage is so full that you can't seem to even turn around. I fear that the only solution is probably money. I'd invest in another hard drive. I tend to stick with 1 TB or 2 TB drives-- I'm always afraid that with a larger drive I will just lose way too much information if it fails!

If your MacBook has a USB-C or Thunderbolt port, I'd get a fast solid state drive if I could-- 1 TB maybe $77 on Amazon (they've gone up!) A slower mechanical drive is maybe $50. If you use a drive just for long-term backup, speed doesn't matter so much, and they cost less. But I've found thumb drives to be pretty sketchy; even two from the same company may be wildly different in quality.

I would reconsider your objection against iCloud Photos. It is the only option to be able to use a Photos Library on different Apple devices with different system version. With my Photos Library in iCloud I can work with my photos on my older macs with macOS 10.14 Mojave as well as on my Macs with Ventura, on my old iPs with iOS 9 as well as on my iPhone and iPad Pro with iOS 16/iPadOS 16. And all are perfectly in sync. i can just grab the device that is near me and access the photos, or use the older software that will no longer run on Ventura to edit my photos on the Mojave Macs. It is the one thing that I really love about Photos.

That's great! People tags are always slower than you expect (or hope.)