Prueba del éxito:

cat and cut

cat prueba_nc.txt | cut -f2 -d'>'
AC_000001 [AC_000001] Ovine adenovirus A, complete genome.
AC_000002 [AC_000002] Bovine adenovirus B, complete genome.
AC_000003 [AC_000003] Canine adenovirus 1, complete genome.

cut

   cut -f2 -d'>' prueba_nc.txt
   AC_000001 [AC_000001] Ovine adenovirus A, complete genome.
   AC_000002 [AC_000002] Bovine adenovirus B, complete genome.
   AC_000003 [AC_000003] Canine adenovirus 1, complete genome.

AWK

awk '{print $2}' FS=">" prueba_nc.txt
AC_000001 [AC_000001] Ovine adenovirus A, complete genome.
AC_000002 [AC_000002] Bovine adenovirus B, complete genome.
AC_000003 [AC_000003] Canine adenovirus 1, complete genome.

So you want to get rid of the filename?

In that case, use the -h/--no-filename option of grep.

Or you also want to get rid of the >? You could pipe the grep to sed, e.g.:

grep -he ">" *.fasta | sed 's/^>//' > prueba_nc.txt

for file in *.fasta; do grep -e '>' | cut -f 2 -d '>' > "`basename .fasta`.headers.txt"; done

Will take each file with .fasta extension in your cwd and grep for the headers, and cut the headers and take everything after the '>' and place them into a file named with the prefix from the original fasta file now with the extension .headers.txt

Using seqkit

seqkit fx2tab -in *.fa > headers.txt