What causes the duration of moon phases (the time the moon is visible above the horizon) to vary for an observer in the northern hemisphere? Why does each phase of the moon stay up a different number of hours per phase, so that, for example, January’s full moon was up twice as long as the new moon? Plus every month this bulge shifts and at some point the opposite phase is doubled. Three June full moon nights takes six March 1st quarter nights. Why does this happen?
The interval between moonrise and moonset varies for the same reason that summer days are longer than winter days.
How long a celestial object appears above the horizon depends on the observer's latitude and the object's declination (angle north or south of the celestial equator).
At full moon on 2019-01-21, the Moon's declination was +20$^\circ$, and there were 15 hours between moonrise and moonset at 40$^\circ$N latitude.
At new moon on 2019-02-04, the Moon's declination was -18$^\circ$, and moonrise and moonset at 40$^\circ$N were only 10h10m apart.

Another way to think of this is that when the Moon is in the first quarter, it's within about 5 degrees of where the Sun would be 3 months later; last quarter, the Moon is where the Sun would be 3 months earlier, etc.
A full Moon in June is up about as long as the Sun in December; a first quarter Moon in March is up about as long as the Sun in June.