What does “Meanwhile, this app’s server code will run on Anvil’s shared servers.” mean?
It means that, if your app is configured in that mode, your Server Module will run on the same (shared) fleet of servers as apps from standard (non-Dedicated) accounts. (Normally, as you have a Dedicated plan, your server code runs on separate resources segregated from everyone else.)
Can I still share tables and dependencies from my dedicated server account?
Your server code will still have access to your Data Tables through the normal APIs, but you will not be able to use Direct SQL access to connect to your database.