"then the two mooring pennant loops are tied together with a string to snug them up to the front of the cleat."
Could be local tradition, or the best way to use what hardware you have. I'm more familiar with two separate mooring lines, each with a loop spliced in the end of it. Each line comes up one side of the bow (over a roller or chafe strip) and then is made fast to a mooring cleat, a conventional horn cleat.
Personally I line to see a cleat that is sized to match the line, so that the loop can be passed under the center of the cleat and then dropped over the horns, spreading the load and the chafe areas, quick and simple to hook up or drop.
If your boat "sails" at the mooring, the two lines can twist up around each other where they come off the mooring ball...but this way you've got two fully separate and redundant ties to separate cleats.
On the mooring ball itself, the lines should go to a strong ring at the top of an iron bar which extends through the mooring ball, with another ring at the bottom where the ground tackle is attached. Inspecting those fittings is critical, once a year may not be enough.
But there are many ways to skin a cat. Mainly, whatever the local tradition is, or harbormaster reqs are, make sure that whatever the mooring is attached to really is adequate for YOUR boat, and in good condition all the way down. Most folks would have a diver (or the mooring company) inspect that once or twice a year to confirm it. And get the results in writing. A worn $10 fitting can cost you a boat.
If this is a small boat and there's just that one bow cleat? Taking a line back to the mast and using it as a second sampson post (strong point) is more than I'd do every day, but a good long-term or storm tactic. The mast also makes a good towing point for small craft that have nothing better available.