As mentioned above: DEVONthink must be installed in the /Applications folder. Is this the case?
Also, does DEVONthink have Full Disk Access enabled in System Preferences > Security & Privacy > Privacy ?
yes and yes, I went through all and any advice that I found here and in other threads, nothing seems to make a difference.
BTW, neither does MDM – my 14" MacBook Pro and my Intel mini are both managed under MDM, and everything works – however, they were installed more than a year ago and updated since then, while my M2 Air and Mac Studio are both installed freshly (Ventura and upgraded to Sonoma since) and both do not have the sharing extension.
Is DEVONthink’s extension enabled or at least shown in System Settings > Extensions > Sharing?
ok, this is the solution:
on Sonoma (and Ventura, I suppose) the new System Settings app has the Extensions subsection in the “Privacy & Security” section.
Once I went to System Settings > Privacy & Security > Extensions > Added Extensions the DEVONthink extension actually shows up and I could enable it there and it shows up in the list of sharing extensions e.g. in Safari – problem solved!
Interestingly, if I now go to Edit Extensions in the Sharing menu, I STILL do not see the DEVONthink extension (will check what happens after I reboot later)!
That unfortunately means I cannot re-order the sharing items and move the DEVONthink sharing up to the top, where it belongs 
Very strange, so this must be a real bug in MacOS, probably since Ventura!
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Thanks for sharing that information, no pun intended 