Amulek, likewise prepared by a vision, knows that Alma is a prophet and is fully ready to receive him and offer him food.
Culture: It seems surprising that Amulek would announce his lineage as Nephite. To understand the implications of Amulek’s declaration, we need to review what it is that he might be saying. First, he may be declaring that he is Nephite as opposed to Lamanite. The Book of Mormon often uses those two terms as polar opposites to divide people into the “us/them” categories. This certainly cannot be the case with Amulek, because he is living in Zarahemla’s polity, which is assuredly Nephite. Thus it would be extremely surprising for Amulek to declare that he was Nephite-not-Lamanite. Alma would have deduced that fact simply from where Amulek lived.
Another possibility is that Amulek is declaring his actual lineage—that he is of Nephi’s literal “tribe.” Certainly Alma’s surest introduction into these cities he visits would be with direct kin. In fact, Amulek later declares his descent from Nephi (Alma 10:3).
The third possibility is that Amulek is declaring his politico-religious orientation. In context, this reason appears to be the most logical. Amulek is living in a city dominated by the order of the Nehors, a conflicting and competing politico-religious order. When Amulek declares that he is a Nephite, he is telling Alma that he espouses the same politico-religious worldview as Alma (Nephite, not Nehorite) and is therefore an ally in a setting where Alma could expect only opposition. For Alma, this declaration of affiliation would have been more important than a recitation of genealogy.