For that all you need is messenger app on your phone and messenger.com on your computer. You don’t need Facebook main app or website.
This might be just specific to very international people like me who travel and work in many different countries so our phone numbers change often and keeping track of that is almost impossible.
I guess this also varies tremendously by region. WhatsApp completely dominates non-work communication in Argentina; we don't use SMS, nor phone calls, and FB is pretty much is second place.
AFAIK, in the US people actually use SMS a lot, and I'm sure it's a bit different everywhere else.
More on topic though; you can use Messenger without an FB account and still find people by username.
It seems that you have phone number that is in some way associated with you.