Changing IP for TimeMachine hard drive

We don't do that with the Time Machine interface. Research your NAS device documentation to determine how to assign it a static IP address. Subsequent to changing that address, you may need to remove that Time Machine destination in Time Machine's Settings, and add it again.

What will happen with my old backups, which I obviously don’t want to delete?

Nothing. After adding that same TM backup destination again, TM will request authorization credentials for the NAS device. Once it connects, a dialog will appear indicating that device already contains a backup for that Mac, asking you if you want to create a new backup or use that existing one. Choose the latter. To reiterate, it will remain unchanged regardless of that choice. You will then be asked for its encryption password. A long time will ensue during which TM examines the existing backup to identify changed items that need to be backed up, after which normal incremental backups will occur as they always have.

Can I simply remove and add the sinology directly without reassigning the IP of it?

Almost certainly... but I am not familiar with that NAS which is the reason I can't provide specific instructions.

Will that change any of what you explained?

No. You may need to "reconnect" to the NAS by removing it from Time Machine's Settings, and then adding that same device again, using that same dialog:

• Open Time Machine's Settings (or Preferences, depending on the particular macOS version), and scroll down to Add or Remove Backup Disk...
• Select the existing disk, Remove, confirm with Stop Using This Disk, then Add or Remove Backup Disk... again, check the Encrypt box, authenticate, and let TM work as it always has.
  • If you have only that one backup disk and no others, the Add or Remove Backup Disk... selection appears as a "Add Backup Disk" button.
  • It's not a good idea to have one and only one backup disk.

None of the above alters the backup's existing contents. It will be preserved until you deliberately erase it.

Ref: Back up your files with Time Machine on Mac - Apple Support

Will it be seen as a completely new device and cause problems?

It shouldn't. You will know you were successful when Time Machine tells you the device already contains backups for that Mac, and asks you if you want to create a new backup or use the existing one. Don't create a new backup, but if you should mistakenly select that choice, TM will not erase the old backups. It will just occupy space on the NAS, and the first new backup will take a long time.

Great! Thanks for the update.