I think you may be looking for the 2004 game Tribes: Vengeance. It's a sequel to the online multiplayer shooter, Starsiege: Tribes, which was itself a sequel to the Metaltech and Starsiege games.
Set hundreds of years before the events of Starsiege: Tribes, Vengeance depicts the birth of the growing Tribal War. It focuses on the events surrounding five different characters over the course of two generations and how they each contribute to the developing war. The story ("The Past") begins with a Phoenix sub-clan leader named Daniel abducting the soon to be Queen, Princess Victoria. He takes her to his home world to show her the injustices done to his people and the two eventually fall in love. During this time, a cybrid assassin named Mercury is hired by an unknown contractor to eliminate Daniel, but the contract is canceled moments before the shot is fired. Eventually, Victoria and Daniel try to make amends between the Imperials and the Phoenix, but it all ends disastrously when the Phoenix's enemies, the Blood Eagle tribe, stage a raid on a Phoenix base disguised as Imperial troops. In rage, Daniel kills the Imperial King, Tiberius, whom Victoria avenges by killing Daniel. It turns out that Victoria is pregnant with Daniel's child, who is born female and named Julia soon thereafter.
The opening is on a starship, but it's fitted out like a medieval keep, and the early gameplay has you going into lower levels of the ship to make your way to the bridge. It does involve different factions, jetpacks, laser weapons, and arena combat.
When you mentioned jetpacks, my first thought was Tribes, which was well known for how central they were to gameplay, which led to me searching for
sci-fi shooter two families jetpacks, which linked to Tribes: Vengeance, from which I found the plot description, after which I looked up the Lets Play to find out whether there was a castle at the beginning.