Oh yes, sometimes it's 'wanting nanny' because she has chocolate biscuits (or a dog to play with, whatever) and you don't. Or sometimes it's 'I'm confused, I thought it was nanny collecting me today'. Sometimes a different colour coat or bag will confuse.
And of course, sometimes it's plain confusion as any reception teacher who is frequently called'mummy' will tell you!
Our little one, same age, asks for 'my potatoes', meaning sweet potato wedges (brother doesn't like them). Then when they are on his plate, he says 'I don't like those'. So I put an ordinary potato wedge next to it, he sees the difference and says 'these are mine they're yum'. But it seems he can't see what they are until he compares them.
Always worth remembering that young children may appear to have good language skills, but don't always quite know how to express themselves! I too think you did absolutely the right thing.
Actually one of my favourite family anecdotes about kids getting different grandmas straight in their mind is about one grandma calling on the other whilst the GC (about 8 I think) was there, and was offered a cup of tea. The GC said 'I wouldn't bother, here you have to wash your hands and sit at the table'!!!!