Antenna - measuring current
1) Yes of course you can measure antenna current. It just takes a suitable ammeter.

Whether this makes sense or not is just a matter of what is your application.

Antenna current is commonly measured in MW/SW high power transmitters, it's mostly done to monitor output power or as an input to automatic antenna tuner. 

Nowadays a simple current transformer is invariably used while in the early radio era this was often a hot wire ammeter.

In other fields, like higher frequencies or lower power or in receiving applications it is much easier to monitor some other quantity instead of antenna current.

2) There's no difference between antenna current and closed circuit current it can simply prove hard or impossible to wire an ammeter somewhere in an antenna without disturbing its behaviour.

E.g. you can easily place a small ammeter halfway one of the brances of a 20m transmitting dipole and read (maybe with a binocular) less current than the feeding point.

But you cannot absolutely do anything on a 20mm cellphone GHz range antenna.