Excuse me for necro-posting, but the declension of nouns in the second line here made me suspicious and I turned to the dictionary. Allen and Greenough's New Latin Grammar (1903) entry on the dative of possession* (para. 373, pp. 232-233, online at https://dcc.dickinson.edu/grammar/latin/dative-possession) states that it is the one for whose sake something exists that is in the dative, e.g. Est mihi domī pater (Ecl. 3.33) -> I (dat.) have a father (nom.) at home, literally there is for me at home a father; est mihi liber -> I (dat.) have a book (nom....
I suppose that is a possible reading, but in my opinion a most unnatural one. Compare: "No dog has the home". This can technically be parsed as "the home [a specific home I'm talking about] has no dog", but this would be a very weird word order in English. Furthermore, if one is making a simple general statement, which is one reading of the OP verse, one is by that token not talking of a specific something, so one does not expect a definite article: "No dog has a home" or "No home has a dog". "The" would be warranted in a didactic or normative text, e.g. "... (read more)