Is it possible to share the same Cisco AnyConnect application for multiple companies with different domains on the same computer?
-You can utilize the AnyConnect client to connect to different sites. Sure the supported versioning comes in play here, but the software itself is not site specific. However, the vpn profiles obtained from the respective sites are. These profiles contain site specific settings. On Windows the profiles belong here: C:\ProgramData\Cisco\Cisco AnyConnect Secure Mobility Client\Profile.
Additionally will each company be able to see the traffic on the other domains if I use the same app?
-No. Each connection will be treated separately which will be based on the profile settings and you initiating a connection to either or via the gui dropdown selection(assuming always on is not in use here at either site).
HTH!