You could say something like الكلب ملكي ("The dog is my property."), but the thing is Arabic is quite fond of repeating words like that when in English it would sound redundant, so translating this as "The dog is mine" flows quite naturally if you consider the rest of the language, you know. The -ii ending, which means "my," needs a noun and can't stand alone, so you have to either repeat the noun or use another noun. That's not the case in English: "mine" can stand alone.
Grew up in Saudi Arabia and, interestingly, the local dialect has a word for it - "haggi" (Note: The letter "qaf" or "q" is pronounced as "g"in many gulf countries).
I mean, it literally might translate to "my property" but it sure is used in the same context as the English word "mine"