The moon rotates around the Earth, as opposed to the Earth rotating around the sun, and at a much faster rate than once per year, so it is more surprising the timing is as close as it is.

Is this why we can still see the moon sometimes in the morning?

Haven't you ever seen the moon in the middle of the day before? Moonrise and set shift by 50 minutes every day and it's in the sky for about 12 hours. So whether it's in the sky during the day or night or some portion thereof will change week to week.

The timing of moon rise and moon set?

I remember the first time I saw the moon during the day. I thought it was absolutely beautiful and the coolest thing at the time!

Not sure what they meant, maybe they were comparing the 13ish orbits of the moon to the 12 months of the year being close?

I'm saying that it shouldn't be in the least bit surprising that the moon sets at a different rate from the sun. The moon is orbiting Earth at one rate, while the Earth is orbiting the sun at a completely different rate. Why would anyone expect them to set at the same speed?

The primary driver of everything* rising and setting is the Earth spinning, so everything rises and sets at approximately that rate. Every object's rise and set may be very slightly different due to its own motion, the moon being one of the most exaggerated but still barely impacted (~50 minutes/day)

So it'd be pretty normal to expect pretty much everything to rise and set at roughly the same rate

* The only real exception is objects in low Earth orbit like the space station but they aren't rising and setting in the same location due to their non-zero orbital inclination. And stuff in the air like planes?