My hard drive is ExFAT. It ran well before updating to macOS 13 ventura.
Perhaps it did, or perhaps there have been issues developing within the library that haven't shown themselves before now. One of the symptoms of having the library on an inappropriately formatted disk are silent issues that accrue over time.
Does it mean that the hard drive needs to be formatted to Mac OS Extended (Journaled)?
Or APFS. Yes. Remember that reformatting the drive will erase it so back up all important data first.
However, I suppose the same problem will appear when the second hard drive is connected?
It's likely, but you may be lucky.
Is there any thing I can do to make it right or stop the 'updating' process?
Quit or Force Quit the app. There is no other way.