The Linux kernel exposes this info now in `/proc/$PID/fdinfo/$FD`. For example:
# grep ^iff: /proc/*/fdinfo/*
/proc/31219/fdinfo/5:iff: tun0
/proc/31235/fdinfo/5:iff: tun1
/proc/31252/fdinfo/5:iff: tun2
/proc/31267/fdinfo/5:iff: tun3
Tested with Debian 5.8.10.