This problem has been going on for a long time. Two years ago, I booked
IAH-HNL-LIH in first class, P fare. The UA.com screen showed first class next to both segments.
The IAH-HNL was on UA and ticketed in the front cabin as advertised.
The HNL-LIH segment on HI was advertised as first but was booked in coach. Calling HI, I found out that this was a coach fare on their airline. UA agent confirmed subsequently.
I ended up cancelling and rebooking separately -- IAH-HNL at UA.com, and the inter-island portion directly with HI. UA was simply not able to sell me the ticket at discounted First class for the complete itinerary.
Takeaway: Be aware of advertised discount first class fares booked at UA.com when the carrier is another airline. It will most likely book in coach on that other carrier.
Perhaps rather than being aware, we should be able to rely on the regulatory agency charged with preventing this sort of misleading behavior.