Greg: Here's what the American Heritage Dictionary says:

mate1 (m³t) n. 1. One of a matched pair. 2. A spouse. 3.a. Either of a pair of animals or birds that associate in order to propagate. b. Either of a pair of animals brought together for breeding. 4.a. A person with whom one is in close association; an associate. b. A good friend or companion. 5. A deck officer on a merchant ship ranking below the master. 6. A U.S. Navy petty officer who is an assistant to a warrant officer. --mate v. mat·ed, mat·ing, mates. --tr. 1. To join closely; pair. 2. To unite in marriage. 3. To pair (animals) for breeding. --intr. 1. To become joined in marriage. 2.a. To be paired for reproducing; breed. b. To copulate.

--but I say that meaning #4 is rarely heard. If you refer to a friend as your "mate" you might be understood correctly, but you might also be misunderstood to mean that the two of you are a "mated pair"! I wouldn't recommend using it among people who are unfamiliar with British or Australian usage.

On the other hand, children often have playmates, and this has no connotation of mating.

But usage might have been different in 1912. Or the song might have been using "mate" in the nautical sense—after all, it is a song about boats.