This is a standard adverb meaning "moreover," "in addition": "Mrs. James is likewise our best hostess." Likewise also means "similarly," "same here," "in the same way." It must be this latter meaning that causes a person to acknowledge an introduction to someone who says "I'm glad to meet you" with the silly comment "Likewise." Why not respond "I am, too" or with the ungrammatical but familiar "Me, too"?
Dictionary of problem words and expressions. Harry Shaw. 1975.