It’s an old ass instrument so it had the name long before tape recorders. As for other kind of recording, Wikipedia says:

The instrument name recorder derives from the Latin recordārī (to call to mind, remember, recollect), by way of Middle-French recorder(before 1349; to remember, to learn by heart, repeat, relate, recite, play music) and its derivative recordeur (c. 1395; one who retells, a minstrel). The association between the various, seemingly-disparate, meanings of recorder can be attributed to the role of the medieval jongleur in learning poems by heart and later reciting them, sometimes with musical accompaniment.