It hasn't got to be a "sad song", for me, lots of songs are happy but make me emotional for the memories attached to them.
Here are a few of mine:
I first heard this a few month after I had went through a miscarriage which as you can guess, really hits me hard still hearing it.
This is emotional for me because it was always "our song" with someone special who is no longer in my life, we fell out badly and haven't spoken since.
Possibly a strange one for most, but this was always my grans favourite song, she is still here but has Dementia so pretty much doesn't remember her life and even her family sometimes.