Up, as a word, is very productive in sports.
From the OED we have two scoring related definitions, though neither of these have the meaning we seek.
12.c. (At) the number or limit agreed upon as the score or game.
13.d. (So many points, etc.) in advance of a competitor.
For 12.c. as applied to tennis see Julian Marshall's The Annals of Tennis. In a discussion of which side is preferable in the game of Long Fives, Marshall calls games scored to eight "eight-up" and games scored to eleven "eleven-up."
This potentially could be the source of using [number] up to indicate a tie: if a game requires two points to win both competitors could be at the agreed upon scoring limit (e.g. both at 11, or 11-up). This would indicate a tie in the last moments of the game, but could have drifted to discuss all ties.
I, though, find Jeff Zeitlin's hypothesis in the question's comments more likely, that the preposition has moved right of the number from "tied up". "Tied up at fifteen" becomes "tied fifteen-up."
In 2007 Christopher Davies wrote that the [number] up usage was American.
The terms fifteen up, meaning fifteen apiece and all tied up, are one used in the US.
While this finding does not necessarily mean the usage is originally American, it did make me wonder if the application in tennis derives from other American sports.
I attempted a search for the first recorded use of any number and up to indicate a tie. Searching "[number] up" and any additional search term related to sports is not helpful because much more common is language that refers to a lead, not a tie.
Instead I searched "tied [number] up" for every number from 1 to 15.
The first earliest result I found was, in fact, from tennis. It comes from a 1937 volume of the Michiganensian:
The meet with O.S.U. at Columbus was stopped short by rain with the score tied, two up.
While this scoring is related to tennis it refers to sets, not points. Whether or not this instance is the first written recording, this term was likely spoken far earlier. Still, this mention indicates that perhaps the usage did not begin with reference to tennis points. This also challenges any (as of yet unpresented) hypothesis regarding the hands of a scoring clock.