They've already had one half-hearted attempt at this about 15 years ago
when Shepherd's Bush(C) was going to become Shepherd's Bush Green. As
part of this plan new enamel line diagrams at Mile End WB carried the
new name, and indeed still do! I'm not sure if it ever appeared/appears
anywhere else and it certainly never got as far as Shepherd's Bush(C)
station itself.

If we can't have two Shepherd's Bush stations where does that leave
Edgware Road, Hammersmith and Paddington?

As for confusion for people arranging to meet outside a station, this
can occur even at a single site where there are multiple exits (e.g.
Victoria).