Well, if you weren't unofficially under-the-table adopted, you could always be kidnapped and passed off as someone's son or daughter. Which -- I don't much recommend. But no, the Houses themselves are closed in terms of prestige, and for a good reason. There is a law in Quuros government that no one of a Noble House can rule, vote, or sway elections, you see. So to perpetrate their relevancy, the Houses have declared monopolies on specific areas. If you need to be healed, you will go to someone trained by House D'Mon, for example. If you wish to purchase a new set of armor, it'd be House D'Talus, and so on. So you could still be an average citizen and work and be of use to the Houses without actually belonging to them at all, and you could actually participate in local government, which I hear is a rightly entertaining shitshow (I say that ironically, as I currently work for a very unpopular Empress).
I very much appreciate the warning, Ms....? I've talked your ear off and haven't asked you anything of yours.