You confuse the truth,
That seem to happen when meddling with things that one has no experience with

I don't understand. I've never done with the database before.
Tbh that was a bit obvious as you also linked to a youtube video that does absolutely nothing with SDF / SQL server compact.
The video only shows that Lazarus is capable of developing database applications in this particular video by connecting to a firebird database.
You wrote:
I've bought a piece of software that stores its database in an .sdf file
You seem to have purchased software that stores its data in a proprietary database format.
According to the wiki the origin of that database format is EOL and has been so for many years (
https://en.wikipedia.org/wiki/SQL_Server_Compact).
Try to see if your software is able to store/save/export the database to another format that /is/ supported by Free Pascal/Lazarus and repeat the quest(ion).
So is it unsolvable through Lazarus?
See paweld's answer.
Personally I would not bother so would advise against it (but I have not a clue about your restrictions/freedoms).