This is too common among oneworld airlines.

BA recognizes my AA Executive Platinum as oneworld Emerald; on the same booking, even when BA MMB link to IB is used Iberia shows me as oneworld Sapphire.

In this instance (LIS-MAD IB marketed and operated), MAD-JFK (BA marketed and IB operated) and JFK-SFO (BA marketed, AA operated) it really makes no difference; all I have to do is show my MADJFK boarding pass at JFK for Flagship Lounge access if I want it; there's no IB F lounge.

But it shows the different PSS' interfaces with each other just don't work very well.