Gothram refers to a lineage in which one is born. When we say son, grandson, great-grandson, etc., we cannot specify more beyond a certain generation; we resort to saying that we belong to a particular gothram which means we belong to that lineage. Vyakhyana sastram provided details on this topic. Gothram refers to the relevant sage’s lineage to which we belongs-to.
All castes/orders have gothram; but they cannot be ascertained through a gothram. Varnam means caste, asrama refers to whether a person is a bachelor, household man or saint, etc. Brahmins belong to certain gothrams and we can ascertain some details based on the gothram.
We cannot pinpoint those who belong to a gothram or not because many belong to a gothram. Asrama refers to the order one lives example family, ascetic, anchorite, hermit, etc.