From everything I have read on here, your Mum thinking that there are two of you is pretty common - it certainly happens with my Mil, who will ask me something for the second time, and when I answer she will tell me that the 'other Ann' said that too. She'll ask for her son, my husband, and I'll tell her he is in the other room - 'No' she will say, 'not your S... - I mean MY S....'. She has even asked my hubby for the phone, so she can 'phone her son and let him know' where she is - he is an only child.
Its dreadfully confusing to deal with, and at times Mil can get upset trying to tell me who the other Ann or S... are, or she will be angry because I can't phone them for her - I just go with trying to change the subject, and distraction - doesn't always (or even, often) work, but when she is in that frame of mind, logic just doesn't come into it, and she can't understand that there is only one of me or him