I wondered if it was possible at some point in the past, but I don't think so. The moon's orbit has been increasing its whole existence. It started ~25,000 km away from Earth [0], and is now at ~384,000 km. That means at some point it was at the distance for a current geosynchronous orbit, ~36,000 km. Unfortunately, when the moon was formed, the Earth's day was only 6 hours long [0], so the geosynchronous orbit was much smaller at ~10,000km (based on [1]), and so I don't think they ever lined up.
It's really hard to break global hydrostatic equilibrium of a planet (in fact, that global roundness[1] is used in the IAU's definition of a planet). Any raising of bumps on the surface will tend to produce depressions elsewhere. For example, Mauna Kea depresses the level of the seabed/crust around it. Likewise, high tide at some points (in the solid earth, the oceans, or the atmosphere) are associated with low tides at other points.
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[1] You could probably toy with models of Earth-moon as a pair of Jacobi ellipsoids or piriforms (pear-shaped, thin ends inwards) but I don't see that working without a much smaller mass ratio and higher spins. Piriform bodies (at least of homogenous self-gravitating fluid, which is a good representation of the mantle) are generally unstable. Maybe that's good if you can find a path that relaxes back to a Maclaurin (oblate) spheroid for the Earth mass that doesn't also relax the (whole of the) "bump", and relaxes the moon to its weak Jacobi (scalene) spheroid.
Really speculating substantially away from what I know: maybe the "synestia" flavour(s) of the giant impact hypothes(e)s for the origin of the moon might be a path to some test simulation codes: coalesce an ellipsoidal (or as I said, piriform or even oviform) moon first and have that drive some aspects of Earth's planetary differentiation (which happens later in that (family of) model(s): <https://en.wikipedia.org/wiki/Synestia>). In particular, the driving should be away from homogeneity in an attempt to escape eventual hydrostatic equilibrium for the Earth-mass which otherwise leaves you stuck with encoding surface features on the (very) thin crust and then dealing with the Mauna Kea problem above. I don't know how you could approach this idea with realistic chemistry though, which I think melts & dissolves this line of thinking.
ETA: Really wild speculation: with unrealistic chemistry, freeze out a long-term solid hourglass structure with the neck at the Earth's centre of mass, piling lots of rocks on the ends terminating just under the surface (but above the mantle) at the poles, and then have one pole always point to the moon Mass. Doesn't at all fit lots of lines of evidence in very old surface rocks, though. Also very hard to wash out tides raised by the sun.
[0] https://physics.stackexchange.com/questions/31429/how-long-w...
[1] https://www.dummies.com/article/academics-the-arts/science/p...