So, the whole point of a vpn is to route all your traffic through the vpn. To websites, all your traffic looks like vpn's traffic. The trust point is to the vpn, because they can only forward your requests if they know it's "you".

This is a good assumption. VPNs do not protect your identity. At best a public VPN will mix your traffic with other users at some exit IP, but it's very hard to trust that the VPN itself is not logging your traffic (or at least the metadata).

The Military might not care about people knowing they are talking, just what they are talking about. In that case, using a VPN makes sense because they can tunnel traffic to a server that is exposed to a private network and use a central chat server there or something. All traffic to that central server would be encrypted.

You'd have to use a technology like TOR to get closer to real anonymous communications on the internet (although even that isn't perfect).