Mohave broke Synology TM backups

Time Machine only officially supports NAS shares over SMB2/3 connections. In a Finder window, upon clicking a NAS share, populate the Connect As… button details including user name and password, or the Connecting… will time out and mount nothing. When you get the authorization info established per the preceding, then the Finder just mounts the Synology share with a single-click, in my experience here.

I have a Synology DiskStation downstairs with the older v5 DSM software, and it connects to Mojave 10.14.5 just fine via the Finder. I am not using the Synology for Time Machine backups, and those mounts are currently AFP over TCP/IP. I have also used SMB2/3 in the past with no issues either.

Finder is also broken.

I think some additional detail would help. What's broken?

Apple's Time Machine software will never be reliable with anything other than a Time Capsule, directly connected hard disk drives, or USB drives directly connected to Apple AirPort Base Stations or Apple AirPort Time Capsules. Complaints regarding other devices have arisen since Time Machine's initial implementation over a decade ago. Search this site and draw your own conclusions from the hundreds, if not thousands, of reports similar to yours.

If you insist upon using a non-Apple NAS device for backups, and you want your backups to be reliable (a desirable feature for backups) then use something other than Time Machine.

Does your drive meets the requirements in this Apple document?

Backup disks you can use with Time Machine - Apple Support