Hi rogertheNo1dodger,
The Update Brain Service (com.apple.MobileSoftwareUpdate.UpdateBrainService) is a legitimate, Apple-provided part of your Mac's internal update system. It needs to talk to the Internet to validate macOS updates with Apple and download any additional resources needed.
If your Macs have the T2 Security Chip or an Apple silicon chip, by default each macOS update is "personalized" for each Mac, and tied to that Mac's exclusive chip identifier (ECID). In other words, the update is tied to the unique serial number for your Apple silicon chip or T2 Security Chip. This is required as part of the "Full Security" secure boot policy.
If you don't want updates to be personalized to your Macs in this way, you can downgrade the secure boot policy to Medium or Reduced Security, which allows for running older macOS versions and installing old macOS updates:
About Startup Security Utility on a Mac with the Apple T2 Security Chip - Apple Support (CA)
Change security settings on the startup disk of a Mac with Apple silicon - Apple Support (CA)
For more info:
Startup Security Utility on a Mac with an Apple T2 Security Chip - Apple Support (CA)
Startup Disk security policy control for a Mac with Apple silicon - Apple Support (CA)