Intrigued by this I looked up the rules which are for last year, but same sponsor.
Rule 3 is a bit long to cut and paste but basically says that the competition shall be marketed under the title 'The Emirates FA Cup' and the club must refer to the title on all materials relating to the competition (list does not include shirts).
The official logo shall have lead sponsor branding.
Images of the cup shall have sponsor branded ribbons.
You have to give the marketing name out over the tannoy and display it wherever possible (not shirts)
Then we get to rule 14 which states:
Rule 14
f) In the Semi Final and Final the participating Clubs will be required to wear sleeve badges branded with The FA Cup Logo unless this is precluded by an agreement with a shirt sponsor who is a competitor of the lead FA Cup sponsor.
So, as Gordyola worked out, we are only displaying it where we have to due to the conflict of interest.