What makes you think it is a satire?
It was most definitely written to confirm the Medici’s biases and stroke some egos.
Yes, that is the opposite of satire.
Did he believe what he wrote in the Prince was noble or just? Probably not, based on his democratic tendencies.
Imagine if you wrote a send-up of Silicon Valley VCs (for example) and tried to be subtly ridiculous about it, but they took your work extremely well and even applied it. Is that not the most successful and hilarious satire of all?