actually it IS operated by West Midlands.
A little class in history and names might be helpful.
Previously to Abellio taking over the London Midland franchise in 2017, London Midland was run as one operation. There was a heritage split between the London commuter traffic worked by what was called Silverlink (and Silverlink Metro) and Central Trains which had the rest of the network before London Midland took over in 2007. London Midland (for which, and its predecessors I proudly worked for many years) tried to bring the two parts together and largely succeeded.
Staff from ex-Silverlink depots (Watford, Bletchley and Northampton), and staff from ex-Central Trains depots (Birmingham New Street, Birmingham Snow Hill, Wolverhampton, Leamington Spa, Stourbridge Junction, Worcester, and the two new depots of Crewe (working all the way to Liverpool and with early hopes of Preston) and Coventry) worked together and routes overlapped, particularly the through service between London Euston and Birmingham New Street via Northampton. Depots in the different regions always overlapped.
When Abellio (initially NS, the Dutch railway and several Japanese railways, now different) took over it was in a political climate where there were expectations that the Birmingham part of the business would be hived off to local government, much as TfL had taken over most of London transportation. So they divided the business into two sections, basically long distance, from Euston to Crewe, and everything else.
They started by calling the whole business West Midlands Trains Ltd, the London, Greater London, Crewe and Liverpool services were called London Northwestern Railway, and the remainder called West Midlands Railway.
Note that the holding company is West Midlands Trains and the Birmingham area services operating company is West Midlands Railway. Confusing, eh?
Whether for laziness or simplification the people running the station displays and National Rail Enquiries have drifted to lumping them together under West Midlands, the holding company. As far as I know that is unique - usually they use the trade name. You don't see Lumo listed as First or even First Lumo. Or Virgin Trains was never called Stagecoach Virgin. But the screens say West Midlands even though the trains are painted either West Midlands or London Northwestern based on their expected routes and maintenance depots (not even going to talk about those - Bletchley TMD RIP) and a train painted one or the other may have either company's employees working it, sometimes both.
So, long story short, out of Euston the crews think of themselves as London Northwestern, the advertising all says London Northwestern, the trains say London Northwestern, the PA announcements all say London Northwestern, and some of the screens and National Rail Enquiries say West Midlands.
hmmph....
So both are right.