My external boot drive denies me admin rights

If you are booting from an external drive, then you are not allowed to modify the root of the file system as it is now a sign & sealed read-only system volume. You may only make modifications to the "/Users" folder. See these two Apple articles about this recent changes one of which began with macOS 10.15, while the other began with macOS 11.x.

About the read-only system volume in macOS Catalina - Apple Support

Signed system volume security in iOS, iPadOS, and macOS - Apple Support

No, you don’t.

All the folders under /Users/(username)

us where you should put your content.

If you insist on putting files and folders at the root level of the drive you can only end in frustration. This is not allowed.

No matter whether there is just one user - you - macOS is a multi-user OS.

Enjoy your mac, and don’t try to fight it. It is protecting you from yourself.

If you want to put data on the root of the drive, then you will need to use another drive freshly formatted for use as a data only drive. If you want the drive to be a bootable macOS drive, then you must live with these new security features and restrictions.

Other options if you want to share data between more than one OS:

• iCloud
• Third party cloud file storage/syncing services
• Local network shared storage (NAS -- like a Synology NAS)
• External USB3 SSD data drive

You probably need to change your permissions settings, then.

1. Turn on your Mac, then press and hold Command (⌘)-R immediately after you see the Apple logo. Your Mac starts up from macOS Recovery.
2. When you're asked to select a user you know the password for, select the user, click Next, then enter their administrator password. 
3. When you see the macOS utilities window, choose Utilities > Startup Security Utility from the menu bar.
4. When you're asked to authenticate, click Enter macOS Password, then choose an administrator account and enter its password.

Then, change your Allowed Boot Media as seen below.

About Startup Security Utility on a Mac with the Apple T2 Security Chip - Apple Support

Booting from An External Drive with Monterey is problematic at best.

This became more evident after and around Monterey 12.3 and above

This does mean it can not be done but . .

Boot from External Drive in Monterey - Apple Community

iMac 2019 5K retina won't boot from SSD drive now that I've updated it to Monterey!!

Cannot upgrade to Monterey with non-apple SSD.

Go to System Preferences->Security & Privacy->Privacy, click on Full Disk Access; click the padlock to authenticate; click + and add Excel to the list. Make sure that the box is checked.

Does that now let you save in the above location?

I wrote incorrectly earlier My Bad😔 and must correct things now.

I wrote " This does mean it can not be done but . . " Incorrect.

Should read >> This does Not mean it can not be done but . . . . and implies it is problematic

Apologies and will be more careful in the future

You cannot gain ownership of the boot drive, nor is there any reason to.

If you explain what you are trying to achieve with this, we can probably guide you.

👍