A little late to the party..
But The Church of Jesus Christ of Latter-Day Saints maintains that you have the actual lineage of that tribe in your blood. Most everyone has multiple tribal bloodlines in them, the one that is chosen is the one which you will receive your blessings from. If you have no blood of any tribe it would be specifically stated that you will receive your blessings through the abrahamic covenant. Those are the ones who are adopted into the covenant.
So yes, it is a literal bloodline
if you have no blood of any tribe, you will be adopted into the abrahamic covenant and receive your blessings through it.
Knowing which tribe you are of is important because it lets you know what your work and purpose in the house of Israel is and who you are in the family of Heavenly Father.
We know the Abrahamic Covenant is the THE birthright of God given to Jesus Christ who will, if we have been worthy and received the endowment/made and kept temple covenants, make us equal partners in that birth right receiving and becoming all that Heavenly Father is. So it makes sense that we have to be of his family/house which is Israel.