You've asked two different questions here, but I'm going to try and answer both of them.
First of all, in regard to how to say "because you are mine", the very literal translation would be something like:
Second of all, let's talk about the difference between "I have" and "there is".
I encourage yourself to picture a Venn diagram for the meanings of "I have" and "there is", and really ask yourself how much they overlap. It may be more than you think.
For example, take this sentence
I have three siblings.
In Japanese this would be
Which translated very literally is
As for me, there are three sisters.
They look very different, but that's because English really likes to stretch out the verb
have. Do you own your sisters? Are they really yours? Or is this just English's way of saying that in your case there happen to be three other people with the same parents as you?
A lot of the cases where we would use
have in English that are not really possessive in Japanese end up being
何々がある・いる, but I think this can be fairly intuitive if you give it a chance.
Lastly, there are a couple other verbs that are used for specific cases.
If you want to talk about owning a pet, you (as user27280 mentioned) use
飼う. Think of this as
keeping an animal.
However, for objects that are explicitly yours, the verb
持つ is also used. It's usually taught as "have/hold", but as you can see here it can also mean "have" in the sense of possession/ownership. You wouldn't use it for your sister(s) or a dog, but you could certainly say
Edit: I also want to draw attention to user27280's answer, and specifically the usage of
私には何々がいる, which implies more possession/relationship than just