The request is for a single WebAppUser table for all the web-sites. The WebAppSession table is a child of WebAppUser so you could always have multiple Session tables (with different names) and use the same WebAppUser table.
However, since you are having multiple WebApps each will have it's own WebAppSessionManager and therefore will create a new Session record in the WebAppSession table on login so a shared WebAppSession table should be no problem either.

It sounds as though you are expecting the one User to run several of the Web-sites in the same browser at the same time. Therein lies a problem since whichever Web-site cookie was the last to be created will control the WebAppSession record found on the next server call, chaos! You will therefore have to do some work in the cookie area. That is not something I have any experience of. Marco might be able to help you. He is rewriting our web-site for us and ran into a similar issue where users want to have the same web-site open to different views in separate tabs. He has resolved that problem now although I don't know how.