Why is the moon's orbit elliptical shaped?

Johannes Kepler published his Laws of Planetary Motion in 1609.

“First law:The orbit of a planet is an ellipse with the Sun at one of the two foci.”

This works for the lunar orbit also; Moon’s path

With minor modifications, it works for the orbits of Jupiter’s satellites and even the orbits of binary stars.

This can be expanded to the two-body problem in space, also called The Kepler Problem

Keeping things simple, these use Newtonian mechanics rather than quantum mechanics and get pretty good approximations. They also use Newtonian gravity, which obeys the inverse square law.

”...a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity.”

The Moon is a satellite of the Earth. A big one, but a satellite nonetheless. A perfect circular orbit for a satellite is only one of an infinite number of a potential orbits.
There polar orbits, geosynchronous orbits, sun synchronous, elliptical, highly elliptical (like many Russian Spy satellites), plus others.
The elliptical orbit of the Moon is one of the factors favoring the Giant Impact Hypothesis for the Moon’s origin..

My favorite orbit (Doesn’t everybody have a favorite?) is Halo orbit. The satellite orbits around the center of mass between two bodies, like the Sun and the Earth, for example. The satellite appears to be orbiting around nothing! It looks like a halo around the smaller body. There are satellites out there at this very minute in that position.

There is also Earth Trailing and Earth Leading orbits. They’re cool too. And of course you can always park something at the Lagrangian points. It’s like putting money in a virtually zero interest savings account. It costs you nothing but it’s there if and when you need it.

A spherical orbit requires a precise speed at the given distance, otherwise the gravitational force is too strong and pulls the object in too much.
needless to say, hitting the right speed has a very low probability for an uncontrolled object like a planet/moon/comet/asteroid

So what we have is a large object caught in the Earth’s gravitational pull and the large object is intent on escape from this attraction.
As it flies across the skies it is caught at the ends of its escape, this causing the moon’s orbit to become elliptical.

@gailcalled has it right. All orbits are elliptical, their characteristics governed by the forces between two bodies. Differences between orbits are partly controlled by their eccentricity, which describes how far the orbit deviates from a perfect circle. That is, a circular orbit is just a special case of the elliptical orbit, in which eccentricity = 0. If the eccentricity value is too high, the object will escape orbit altogether.

I’m so impressed so many of you know so much about this. It’s a bit over my head. Thank you!

It’s important to realize that the earth and the moon actually orbit one another. The moon is a satellite of the earth but it is so massive in comparison to the size of the earth that it would be fair to refer to the 2 bodies as a double planet. We may speak of the moon as orbiting the earth, but the fact is that they are both in orbit around the center of mass of the two objects. And as @dappled_leaves says all orbits are elliptical since a circle is just an ellipse where the 2 focal points happen to coincide. A perfectly circular orbit involving celestial objects is all but an impossibility for more reasons than you can shake a stick at.