The exported list is temporary until you save it. Four taps and it stays in your library — here is exactly where to tap.
After tapping the export button on Anisongs, the first track opens big. Drag upward from the "Untitled List" label at the bottom.
You will see the whole list ("Untitled List"). The Save button sits at its top right.
A dialog asks where to save the songs. Create a new playlist — or pick one of your existing playlists to add the songs there.
Give the playlist any name you like and tap Create. Done — it is now in your library.
The playlist now lives in the Library tab of YouTube Music. Downloads and edits work like any other playlist.