DEVONthink Server does not run on or is related to Synology NAS (except of course if DEVONthink connected to the server is pointing to indexed files on the NAS but I don’t know how these indexed files appear to server users … others can comment). And yes, if you setup the network (and its security) DEVONthink Server can be accessible from outside your local network. For the latter, I must say do that carefully and with great skill.
As mentioned, the Server edition can sync (as can all editions of DEVONthink) but this has no bearing on the use of the NAS for the built-in webserver.
Also, your database can’t be directly accessed on other platforms via syncing. Our sync engine is not like iCloud or Dropbox, etc. It’s not doing single file syncing nor is it providing an accessible copy of your database for you to open or get into. It’s raw, chunked, and optionally encrypted sync data only useful to DEVONthink and DEVONthink To Go.
The Server edition has a built-in web server allowing you to broadcast a database on a network. This allows you to access specific databases in a web browser, including on other platforms. (See the Help referenced below regarding access over the Internet.) It has per-database, per-user permissions you can set to control access to the databases you have open.
More discussion of web sharing is covered in Help > Documentation > In & Out > Web sharing, Windows > Web Sharing, and Preferences > Server.