It really depends on your logging goals. The shiny session token is internal, and links an active browser session to an R process. Also be aware that the token attribute of the session object isn't formal part of shiny's "exposed" API, so it may be subject to change.
If you are interested in logging at a user level, or aggregating across sessions, than this token may not be useful. You also should be careful as there are "reconnect" cases where the same session will reconnect with a different token.
But aside from those caveats, its probably ok to use the token as an identifier for when you server function is called.
In case you are interested, there is documented and exposed client data as well: https://shiny.rstudio.com/articles/client-data.html
And in authenticated sessions on RStudio Connect you'll also see
session$user and in some cases