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.