Yes, they do, I bought one for my Birdy Titanium that wasn't factory equipped with lights.

Now, the fork of the several Birdy models, Birdy 1, Birdy 2 and Birdy 3 are different, there is a different one for Birdy <2016 but is it usable for a Birdy 1 (before 2016 there were the Birdy 1, Birdy 2 hydroformed frame and Birdy 2 World with frame made of tubes like for the Birdy 1 but a different one) ?

The one for Birdy <2016 (sorry, its a Dutch webshop).
The one for Birdy from 2016 = Birdy 3 (same Dutch webshop).