That's not quite right. Ealasaid is the subject of the sentence in English. The button is the subject in Gaelic.
Tha putan aice [There] is [a] button at her
Tha putan aige [There] is [a] button at him
Tha putan aig Ealasaid [There] is [a] button at Elizabeth
So we simply have different words for 'at her', 'at him', 'at'. D
Normally you would use air for wearing, so you that would mean 'Elizabeth is wearing a button'. But I am not sure you can 'wear a button'?
However, air and aig are only one letter different. That should have been marked as a typo, but I think it does occasionally miss typos..