It's actually the same structure as in Italian: il suo cane whereas in French you would say: son chien and in Spanish: su perro Some languages even have both possibilities, like Norwegian: hunden hans/hennes hans/hennes hund
Conclusion - some languages, when expressing possession, use a definite (in Romanian called articulate) article, others an indefinite (inarticulate) article.
in French we would say : j'ai TON chien or j''ai VOTRE chien .depending on whether you are friend with this person or not. In Spanish we say the same : tengo TU perro or tengo SU perro.
and don't look at how many lessons I have in each language. French is my mother-tongue and I live in South America since 1992 after having visited it constantly since 1966 for business. So Spanish has become a mother-tongue to me too. Same is true for Italian, German and Portuguese which I am just "learning" here for fun and to see how DUO is teaching these languages.
But you have to use the definite article (like saying "the dog") because is not "a dog".... it is "THE dog of yours". It's like saying "I have the dog of yours". You can use it in spanish in the same way: "Yo tengo el perro tuyo" but it's more common to use "yo tengo tu perro", like in the english way.
does that mean that if someone had multiple dogs and you weren't referring to any particular one of them, you would be able to say câine?
No, the noun always has a definite article when a possessive pronoun follows.
Apparently "your" implies a specific dog, but Romanian demands using the articulate form of the noun as well in that case. That means -le would not be just a postponed definite article (resembling French le), it is rather an integral part of a word that is used whereever a specific object is meant.
I like your logic, but it is not really like that. The definite and indefinite articles are just ”tools” that help words to form/change a sentence and to give it a specific meaning. Both definite and indefinite articles are used to point out a specific object. It depends on what you want to say.
The syntax is the below part of the ”iceberg” called ”Romanian grammar”. But don't worry, it is not rocket-science. Just don't make theories or compare it to other languages. It will just confuse you. Romanian has similarities with majority of European languages (also Latin, btw), but it has its unique features as well.