It seems that the furnace does not support copy-pasting. When the factory is placed not by the player but by copy-pasting, even if modules are inserted, they still cannot be used. However, if the player places the furnace and inserts modules, it can be recognized. Is this my issue?

Alright, it's indeed my issue. I have a mod that automatically constructs ghost blueprints without the need for construction robots, which results in the structures not being recognized as player-built. I hope my experience can be helpful to you

Hello,
Yes, your guess is correct. My mod "watches" for a limited number of events so when some other mod just places something my mod does not know anything about it.