Does the lineage indicated in a patriarchal blessing mean an actual temporal lineage or does that refer to a lineage of spiritual nature (meaning our responsibilities of being born in the Church)?
Either way, the result is the same. In other words, whether you are a natural descendant in one of the tribes or you are adopted into one of the tribes, either way you inherit all the blessings and privileges of that particular tribe. Take a look at the following quotes:
Encyclopedia of Mormonism, Vol.3
"An essential part of a patriarchal blessing is a declaration of lineage. The patriarch seeks inspiration to specify the dominant family line that leads back to Abraham. The majority of modern blessings have designated Ephraim or Manasseh as the main link in this tracing, but others of every tribe of Israel have also been named. Whether this is a pronouncement of blood inheritance or of adoption does not matter (see Abr. 2:10). It is seen as the line and legacy through which one's blessings are transmitted. Thus the blessings 'of Abraham, Isaac and Jacob' are conferred.
"In addition, as the patriarch seeks the spirit he may be moved to give admonitions, promises, and assurances. Individual traits of personality and strengths and weaknesses may be mentioned. Against the backdrop of the prophetic anticipation of world events, individual roles and callings may be named. One's spiritual gifts, talents, skills, and potentials may be specified with their associated obligations of gratitude and dedication. Karl G. Maeser described these blessings as 'paragraphs from the book of one's possibilities' (Alma P. Burton, Karl G. Maeser: Mormon Educator, p. 82 [Salt Lake City, 1953]).
"It is continually taught in the Church that the fulfillment of patriarchal blessings, as of all divine promises, is conditioned on the faith and works of the individual. Typically, blessings close with such a statement as, 'I pronounce these blessings upon your head according to your faith and your diligence in keeping the commandments of the Lord.' "
Pres. Joseph Fielding Smith Jr.
Doctrines of Salvation, Vol.3, p.171
"If a patriarch gives a blessing and the lineage is not designated, is it considered a patriarchal blessing? Yes. However, a patriarch has the right of discernment to designate the lineage. If in a blessing the lineage is not given, the person may go to the patriarch and ask for that information, and if the patriarch is so inspired, he may declare it and have it added to the blessing. If the patriarch is deceased, another patriarch could add that part to the blessing."