Jack, the only real similarity I see is in the first line, and without any other corresponding features, that may well be coincidental. After all, in the song the girl is "standing on the bridge"; in the poem, a narrator "stands on the bridge."

The meter may not mean much either: many songs use it, including, just for example, "Rolling Home."

Here's a 1902 recording of the poem set to music:

https://www.youtube.com/watch?v=4LvnK5TRM-Q

The tune is unlike that of the song, but the date may be consistent with the song's appearance.