Is a quasi-moon like a quasi -planet (i.e., Pluto)?
Nope. Pluto's designation is based on it's size, mostly. The category "planet", like all categories, is made by humans to conveniently describe the universe to ourselves, and the precise boundaries are constrained (but not determined) by how the universe actually is and how we actually are. Within those constraints we can put the boundary where we like, and in the case of planets, smaller bodies that don't dominate their gravitational neighbourhood have been deemed to fall outside the human-created category we use the word "planet" to label.
Quasi-moons are bodies in solar orbits that have interacted with their quasi-primary such that they are "station keeping" with it. A body like this one will wander around in the general vicinity of Earth as both Earth and quasi-moon travel around the sun together. So from the perspective of an observer on Earth, the quasi-moon executes periodic but non-orbital motion: it wanders in a closed configuration that does not describe a path that goes around the Earth.
This is, like many such distinctions, fairly arbitrary: the sun's gravity at the orbit of the Moon is a good deal stronger than the Earth's gravity at the orbit of the Moon, so one could describe the Moon as being in orbit about the sun, with it's orbit perturbed into a wobble by the nearby Earth. That is, from an outside observer's perspective, the Moon's motion is never retrograde with respect to it's mutual orbit with Earth around the sun.
Consider the view:
O o .
where the O is the sun, the o is the Earth and the . is the Moon. In the configuration shown (with the Moon on the outer wobble of its orbit about the Sun) it is moving faster than average (imagine the Earth and Moon both rotation clockwise around the Sun, and the Moon moving clock-wise around the Earth, so when in the image above it is moving "down" on the page).
But in the situation that obtains two weeks later:
O . o
where the Moon on the inner wobble of its orbit about the Sun, it is still moving "down" on the page relative to the Sun even though it is moving "up" on the page from the perspective of an observer on Earth.
Another way to see this is to consider that the Moon executes a wobble like this once a month, traveling 2*pi*0.25 million miles (lunar orbit is about 250 thousand miles), but at the same time moves 2*pi*96/12 million miles in its orbit around the Sun (which is 96 million miles from Earth), and since 96/12 > 0.25 it should be clear that the Moon's orbital velocity around the Sun is higher than it's orbital velocity around the Earth. Ergo: no retrograde motion for the Moon!
All of this is a very long-winded way of saying: how we classify Moons vs quasi-moons is useful, but--as with all the ways we as knowing subjects classify the objectively real universe we live in--somewhat arbitrary. We could--but don't, so far as I know--have a name for the class of moon-like objects that have orbital velocities around their primary that are greater than their orbital velocity around their primary's primary (most Earth-orbiting satellites fall into this category.) Instead, we have a name for objects that don't execute motions relative to their (quasi-)primary that look like a loop around it from the perspective of an observer on the primary's surface.