In forms like Claudio's house or Claudio's dogs, are there languages in which the Claudio's would change depending on gender and number of the houses or dogs?
In Romani, the genitive case marker looks something like -kVrV or -gVrV, where V depends on the gender (and I believe number) of the object. So for instance, you have
where kher is neuter and vudar is masculine.
In Hindi and Urdu, the genitive particle is ka (masc), ki (fem), agreeing with the possessed.
In most Grassfields Bantu languages of Cameroon, there is an associative morpheme which agrees with the possessed noun in a genitive construction. In the most minimal type of agreement, the associative marker is a floating low tone for noun classes 1 and 9, and a floating high tone otherwise. Noni, a Beboid language, has eighteen different genitive markers, depending on the noun class of the possessed noun (Hyman 1981: 19).
A mixed case is Nkwen, of the Ngemba group of Grassfields, which has a segmental genitive marker for classes 2, 5, 6 and 19 (agreeing with the possessed), a floating low tone genitive marker for classes 1 and 9, and a floating high tone marker for other classes. (Ncheafor 2002)
Most Slavic languages have such inflection form. It applies both nouns and pronouns and both possessor and possessed inflect in agreement.
My samples would be in Ukrainian.
Pron. 1st: Я
"My son" (nominative/genitive/dative, other cases skipped):
"My daughter" (nom./gen./dat.):
Plural form: "my sons and daughters":
Nouns (possessor) are inflected in a similar way:
"Ivan's son" and "Ivan's sons" (nom./gen./dat.):
"Ivan's daughter" (nom./gen./dat.):
There's yet another way of forming possessive by placing the possessor at the end. This way, only possessed inflect while the possessor remains in genitive case only:
син Івана / сина Івана / сину Івана
Your question doesn't specify whether you are interested just in nouns or also in pronouns or adjectives.
In Spanish these forms of the possessive/genitive adjectives/pronouns (terminology depends on analysis/tradition) are inflected for number, and some also for gender, to agree with the possessed rather than with the possessor:
This is common among Romance languages.