I suspect that your NAS is connected using Apple Filing Protocol (AFP). AFP has been fully deprecated by Apple and while that protocol works with macOS 26 Tahoe, it will no longer be supported by Apple starting with macOS 27.
Server Message Block (SMB) is the modern replacement for AFP. Apple stopped supporting AFP on the server side many years ago (with macOS 11 Big Sur, in 2020) but left the client side support going until macOS 27.
Synology supports AFP, but if your NAS is configured that way then a Mac running Golden Gate will not be able to access it (this is the same reason that anyone with a Time Capsule who upgrades to macOS 27 now has an aesthetically-pleasing paperweight).
Turn off AFP on Synology NAS:
- Access the NAS via a browser
- Open the Control Panel
- Select File Services
- On the AFP tab, make sure the Enable AFP service is UNchecked.

Synology's setup instructions for Time Machine backups over SMB3 are here:
https://kb.synology.com/en-us/DSM/tutorial/How_to_back_up_files_from_Mac_to_Synology_NAS_with_Time_Machine
If your NAS is used for TM then ensure that your NAS is configured based on those instructions. If it’s only a file server, you really just need the SMB setup parts of that.
Personally, I use a Synology DS223j with a pair of 10 TB HDDs installed to back up 4 Macs in the house with Time Machine (separate shared folder for each Mac sized at ~2.5x the internal storage), with the remaining space used as general file storage for everyone in the house. Two of the Macs are on macOS 27.0.1 and continue to back up normally to the NAS (and can access the volume set up as a file share normally, as well), so I suspect your problem is due to your configuration settings on the Synology side no longer being supported by macOS.