It's a romantic notion. But no. Wild geese are, for the most part, monogamous, but they have been known to re-mate if their former mate dies. (I don't know if domesticated geese behave the same or not.)
There isn't (as far as we can tell) a specific song (and describing the noise made by geese as a "song" is being quite generous) that serves to identify a specific goose, so the notion of "its mate's song" is pretty much meaningless anyway.
They sing to find their mate
yes they can they sing to get mates and to find their mates
they sing
Maverick and Goose sing You've Lost that Loving Feeling - The Righteous Brothers
Birds sing in the spring to attract mates.
Great Balls of Fire
They sing
no some humans do
yes i beleve so
to attract its mate
they sing to each other
It is still a mystery to why humpback whales sing. One theory is that it could attract a possible mate. IT IS CERTAINLY A MYSTERY ON WHAT THEY SING ABOUT!