No, the Byzantine Rite is used by several Eastern Catholic Churches, and several orthodox.
Sure be we are still particular Churches not just rites.
Ruthenian Catholic Church uses Eastern Rite, particulary Byzantine Rite of worship, theology and many more things I am not able to name right now.
Eastern Catholic Churches all have hierarchy of their own, hence they are particular Churches. They are all using Eastern Rites BUT in theory if ever one Eastern Church changed their practice of worship or theology, other Churches would be unaffected. Same way Latin Church is separate from them, they are separate from each other. Fact they use similar (or same) Rites means little in organization.
I’m Latin, but I love the Byzantine Catholic Church. I go two weeks to Latin Mass, then one week to Divine Liturgy. But on all Holy Days of Obligation I go to Latin Church. But I’m not Byzantine, just Latin.
No theory there–there are differences in liturgy among the different Byzantine churches.
No, you’d only belong to one sui iuris church.
But, you can be involved in both, if you’d like (I am).
I am canonically a Latin, and my home parish is Latin, but I love my Melkite friends, and they have my support.