The pieces on the Egyptian checkerboard were called 'dogs'. The 'dog watch' probably is derived from the same concept. Wikipedia again:

The name is said to derive from Sirius, the 'Dog Star', on the claim that Sirius the first star to come into view on the first dog watch. In fact, however, the time of the rising of Sirius varies with the time of year, so that for six months of the year it is below the horizon at sunset. Moreover, the first dog watch (1600–1800) is typically in daylight, and the stars are not visible.

An alternative folk etymology is that the name comes from the fact that someone tasked with one of these 'half' watches was said to be 'dodging the watch', taking or standing the 'dodge watch'. This became shortened to 'dog watch'. Another variation is that those sleeping get only 'dog sleep' in this watch. Stephen Maturin of Patrick O'Brian's Aubrey-Maturin series makes the pun that the dog watch is so-named because it is 'cur-tailed' ('curtailed', i.e. shortened).

The reason behind this watch's existence is that in order for the crew to rotate through all the watches it was necessary to split one of the watches in half, to create an odd number of watches in a ship's day. This allowed the sailors to stand different watches instead of one team being forced to stand the mid-watch every night. The choice of time also allows both watches, if there are only two, to eat an evening meal at about the traditional time.

One watch is cut short, creating a single (non-paired) element. This happens after 9 double-hours counted from midnight.

And it is time to eat. 4 bells are striking. It is a half-way station.

At New Year there were 16 bells striking (equal to 472 / 29.5). A year contains 4 quarters, 2 for the 'new' year and 2 for the 'old' year.

24 hours are divided into 6 times 4 hours. There are 3 watches for the 'new year', but the dog watch causes the 'old year' to be 4 watches. 3 + 4 = 7 as the number of planets: