Literal or Figurative -- That is Often the Question
The first and most general observation about this word is to note that English words ending in the suffix -ship (a Germanic suffix, related to -schaft in German) are often not translated into one word in Spanish. Those that are, such as friendship -- amistad -- often fall into that category of Spanish nouns ending in -tad or -tud (and are, by the way, all grammatically feminine).
The concept of ownership, when speaking literally of property, is not expressed by one noun that refers to the abstraction of ownership. The nouns propiedad (property) and dueño/a (owner) are enlisted in phrases to show that something, such as a house, is the property of so-and-so or that so-and-so is the owner of the house.
Juan es dueño de esta casa (Juan is the owner of this house).
Esta casa es propiedad de Juan (This house is Juan's property).
To ask about ownership -- ¿De quién es/son...? is the most common expression. Note the following examples of how this expression is used:
¿De quién es este terreno? (Whose land is this?)
¿De quién son estas herramientas? (Whose tools are these?)
If the one asking expects there to be more than one owner, the plural of the interrogative pronoun must be used. Note that the thing owned may still be singular:
¿De quiénes es esta casa? (Whose house is this? or more clearly expecting a plural answer in English: Who are the owners of this house?)
The word cuyo and its derivatives (cuya, cuyos and cuyas) all mean whose. Note how it agrees in gender and number with the thing owned, not the owner!
El hombre, cuya casa está en la colina, acaba de comprar este terreno (The man whose house is on the hill, just bought this parcel).
When the word ownership is used as a verb in English with its focus on property ownership, the verb for taking ownership is adueñarse or apoderarse (to take possession). However, when the verb implies taking responsibility, stepping up to the plate in some sense, committing oneself to something, the verbs to use in such cases are: responsabilizarse, encargarse, comprometerse.