Understand the post two above ... but "fare class" refers to the first letter in the fare basis code. It's an H-fare. Even if AA wants to tell you it's a 'B' fare. Lots of airlines put additional letter in the fare basis code, but those refer to the fare family and not the fare class.
The booking class is what really matters for earnings, changeability, services, upgrades, etc. So simply calling it an "H fare class" is generally going to be very confusing without also specifying the booking class. It's a B fare class for the things people care about and there are no Basic Economy B fares that actually start with a B in the fare basis code. You could also call it a B fare with an H "fare basis" if you wanted to indicate the underlying coach class used for the dual inventory check.
This is also true with domestic FC fares. If you have a VUAHZNR1 FC fare, calling it a "V fare class" would imply it's a coach fare (with no checked baggage allowance). The reality it is truly a FC fare fare and it is perfectly correct to call it an R fare class (the booking class) FC fare with a 2 comp checked bag entitlement. The only domestic FC fares that do not start with a coach class in the fare basis code are full J fares. You shouldn't accept a downgrade in IRROPs just because you didn't buy a full fare JA2AA FC fare for $2K or more.