You have the right approach, but you need to fieldsplit on whitespace, not commas (`-F,`). I'm interpreting your question as "the lines should be saved to files with names equalling that in column 1".
$ cat file.txt
BB1520_v1.0_0103 exon 3329 3601
BB1520_v1.0_0103 exon 3650 4040
BB1520_v1.0_0089 exon 1 862
BB1520_v1.0_0089 exon 1312 1357
BB1520_v1.0_0089 exon 1423 1531
BB1520_v1.0_0084 exon 5196 5283
$ awk '{ print >$1".txt"}' file.txt
$ cat BB1520_v1.0_0103.txt
BB1520_v1.0_0103 exon 3329 3601
BB1520_v1.0_0103 exon 3650 4040