I have a bird seeder right outside my computer room window, and I've fed birds for over a decade now. There is a flower garden right outside this window. I'm a person who's very curious about weeds, and I look them up in a weed book or on the internet so I can fight it in the best way possible. I've yet to find a foreign weed in that garden, and I've used all types of bird seed, from the expensive to the inexpensive.
I have very little problem with weeds there. Plenty of seed falls out of the bird feeder, too--so much that the ground feeders, like the mourning doves, just feed off the ground.
Also, birds eat weed seeds, too.
I'm wondering if the bird feeder might attract some migratory birds who stop for lunch on their way to wherever. While they're at lunch, they also stop to use the restroom, so to speak, and might be leaving you some weed seeds from another area.
If you're up for it, take a pic of the weed and let's see if it can be identified.
Also, next time I talk to my aunt, I will ask her about this. She's been a bird feeder and gardener for 60 years, so she might have some input, too.

Cher