Does the use of "port" for one side of ship relate to the word "port" meaning harbour?

In this travel.se question, the words port and starboard are mentioned, and one answer says:

I would say it is convention that ships dock such that the Port is on the left, from which the term for the side gets its name ("The side of where the Port is").

That never occurred to me before. Is there any basis to it?

Excerpt from trivia on the Navy website:

Port and starboard Port and starboard are shipboard terms for left and right, respectively. Confusing those two could cause a ship wreck. In Old England, the starboard was the steering paddle or rudder, and ships were always steered from the right side on the back of the vessel. Larboard referred to the left side, the side on which the ship was loaded. So how did larboard become port? Shouted over the noise of the wind and the waves, larboard and starboard sounded too much alike. The word port means the opening in the "left" side of the ship from which cargo was unloaded. Sailors eventually started using the term to refer to that side of the ship. Use of the term "port" was officially adopted by the U.S. Navy by General Order, 18 February 1846.

And the United States Navy order by George Bancroft:

GENERAL ORDER UNITED STATES NAVY DEPARTMENT,

Washington, February 18, 1846

It having been repeatedly represented to the Department, that confusion arises from the use of the words "Larboard" and "Starboard," in consequence of the similiarity of sound, the word "Port" is hereafter to be substituted for "Larboard."

GEORGE BANCROFT

The OED says it comes either from port (harbour) or port (gate), but whichever it comes from, the explanation is the same.

Etymology: Either < port n.3 [A gate, a gateway; spec. (from the 14th cent.) that of a city or walled town. Now Sc. (hist. exc. in the names of the spot or street where a city gateway stands or stood formerly, e.g. West Port in Edinburgh).], or < port n.1 [A town or place possessing a harbour which boats use for loading or unloading, or which forms the starting point or destination of a voyage; spec. such a place where charges may be levied under statute or by prescription on boats making use of the facilities. Now also occas.: an inland port.].

Whether derivation from port n.3 or from port n.1 is accepted, the underlying explanation is probably broadly the same. When the steering apparatus was on the right side of the vessel (compare starboard n.), the vessel when in port (compare port n.1 3 [A place on a coast or shore which boats use to shelter from storms, or to load and unload; a harbour, a haven]) would normally be placed so as to lie with her left side alongside the quay. Any opening to allow entry or loading (compare port n.3 2 [Naut. An opening in the side of a ship for entrance and exit, or for the loading and unloading of cargo; (also) an aperture in the side of a ship for a cannon; a porthole.]) would also have to be on this side.

Derivation < port n.3 appears the more likely of the two possibilities. This side of the ship was probably already known as the ‘loading’ side of the ship (see larboard n.) and as such contrasted with the ‘steering’ side (starboard n.); it is likely that these associations were maintained. Furthermore, port n.1 is not normally used spec. for the quay or a similar structure.

A suggested derivation < French bâbord (see babord n.) is not likely on phonological grounds and is not supported by evidence.