You don't need a separate authenticator app for each device, the one on your phone will work for any number of client devices, since the items it's authenticating are accounts related to you and not the devices themselves.
That's why Microsoft's authenticator only works on phones, since that's a device you typically take with you wherever you go, allowing it to work not only for your Microsoft Personal account at home and Microsoft Work or School account when there, but also the opposite, since typically you might need those accounts in these different locations as well.
Rob