Star Trek: 10 Secrets Of The La Sirena
8. Yiddish Planetary Union
A wealth of supplemental information (including everything you could possibly want to know about Ganmadan, Freecloud, and Thad Riker-Troi's fictional Ardani), Star Trek: Picard creator Michael Chabon stated on Instagram last year that the La Sirena was manufactured at the Kaplan Shifbau on the Hatzeplatz colony world.
Don't recognize that one? Well it's actually a reference to Chabon's 2002 novel, The Yiddish Policemen's Union which was originally titled "Hatzeplatz".
Beyond that, the origin of the La Sirena has yet to be explored on screen, though designer Mark Yang described the history of the ship from the perspective of the Star Trek: Picard art department:
By the time of Picard, it wasn't brand new. I think it was probably constructed a little after the Voyager era and then changed. The concept was that it was used for Starfleet adjacent operations. Someone had been using it to do things it wasn't originally meant to do, so every few years it would get enhanced equipment. It would have been a civilian cargo cruiser that they added things to over time. Then someone got a hold of the ship and started to bring everything together, putting in more Starfleet equipment.