That is odd, but the TS statement appears to be correct. As a Project Manager, I can share the file using either UI. As a Project Member, I cannot share via RMB, just as you have shown.
If you attempt Add to Project using the Actions menu, can you complete the process? If you cannot, then the TSE is correct.
PTC does do some additional processing of the RMB Actions compared to the Actions menu in the toolbar. All the RMB actions are validated based on your account permissions where the Actions button in the toolbar is not.
Either way, it should be consistent.
-Scott