Hello Shoiwhoathoaph67
The RE80 is a TILO train, a joint-venture of the two railway operators SBB and Trenord.
In Switzerland, the train operator is irrelevant. Here, we have trains from different countries (ICE from Germany, Railjet from Austria, TGV from France, Astoro from Italy, etc).
As soon as they join the Swiss rail road network (in your case at Chiasso) Swiss train tickets are even valid on those. This is the truth of the great success of our public transit system: one ticket fits all.
Kind regards,
Roland