Since none of Putin's two daughters seem to want to play any prominent political role, a short-lived House of Putin would most likely be succeeded by a House of Shamalov, as his son-in-law Kirill Nikolayevich Shamalov, a jurist born in 1982 as son of Putin's old Petersburg cronies, became Russia's youngest billionaire thanks to his father's and father-in-law's influence, based on control of the banks Bank Rossiya (his father's) and Gazprombank and the petrochemical giant Sibur. A pure product of oligarchic nepotism.
The man looks like an anti-Semitic caricature (google him!), but despite its sound the surname Shamalov is apparently geuinely and ethnically Russian, but based on various Turkic first names allegedly given as mocking nicknames to children of small stature! So he is nearly the same age and of similar outlandish appearance and oligarchic tendencies as Grand Duke Georgiy Mikhailovich.
Putin's other son-in-law, the mysterious but apparently handsome Dutch businessman Jorrit Joost Faassen (born in 1980), is much less prominent and of course a foreigner, though no doubt Peter the Great would have smiled in his grave if a Dutchman was to sit on his throne!
BTW only one of Putin's daughters, Mariya, is called Putina. The one married to Shamalov, Yekaterina, has taken her maternal grandmother's patronymic and is called Yekaterina Vladimirovna Tikhonova.