Apparently it means that in the minds of some recent song writers. These lyrics could be the result of historical separation from being a society of farmers and riders of horses. I remember in Sunday school that I was taught what a manger is. I wasn't a farmer, but I still had an aunt and uncle who were. My grandparents raised a bovine when my mother was a child, and my father rode in a horse and buggy as a child. Maybe people don't know because manger scenes are taken down from public places.
Manger is taken from the French word, dmanger, or, to eat. Just as in Spanish it is comer, German essen, Hebrew ochel, etc.
For some odd etymological reasopn, or linguistic if you prefer, people borrowed the word from French in English to depict a barn or a trough for feed. Jesus was born in a manger, and is said to have first slept in the feeding trough or manger, on hay. This is mostly tradition of men but based on the reality of His circumstances at birth.
It is not really so much a dogma or doctrine as it is simply a harmless tradition.
There is nothing at fault to call his birthplace a manger if it was a barn because in those days it was not just for protection but for feeding, just as barns are today.