This answer is based on what I remember from Schuller's General Relativity course, where he also mentions Newtonian spacetime in these terms. I'm not fully sure if it is identical to what he is doing in the Mechanics course (although I'm confident on it) because I don't speak German, so I didn't watch the course.
Yes, there are many Newtonian spacetimes. It should be noted that what he is calling a Newtonian spacetime is not Galilean spacetime, which is the usual spacetime of Newtonian mechanics used to solve, e.g., a harmonic oscillator. Schuller's Newtonian spacetime incorporates Newtonian gravity, so you do have multiple different Newtonian spacetimes. In fact, each solution for the gravitational field leads to a different one. A spherically symmetric Newtonian gravitational field leads to a Newtonian spacetime, while a uniform Newtonian gravitational field leads to another, completely different one.
As for the choice of underlying set $M$ and $\mathcal{O}$, I guess the answer is in your original statement: the spacetime is defined by five properties. These are pretty much up to you and you should choose an option that models correctly what you're expecting. If you're familiar with General Relativity, you'll notice a similar "problem" occurs in there. Suppose you pick a spacetime $(M,\eta)$, where $M$ is some so-far unspecified differentiable manifold and $\eta$ is a flat metric. While a possible option for $M$ is $M = \mathbb{R}^4$ (yielding Minkowski spacetime), $M = \mathbb{T}^4$ (a 4-torus) is also possible. How do you know which one is correct? Depends on what you're doing and the conditions of the physical problem. In General Relativity, it is conjectured that it is impossible to actually probe the topology of spacetime (a result known as topological censorship conjecture).
Questions and points raised in the comments
Doesn't Hamiltonian mechanics already deal with constraints?
The manifold meant here is somewhat different. In usual Hamiltonian mechanics, gravity is understood as a force. In here, it is understood as spacetime curvature (although not in a metric sense). We're not imposing different structures on the configuration space, but on spacetime itself.
Shouldn't spacetime be restricted by orbital stability? No. In General Relativity, for example, we use the word "spacetime" to refer to different physical situations, not the actual physical and incredibly complicated spacetime in which we actually live in. Schwarzschild spacetime models a spherically symmetric compact object. FLRW spacetime models an isotropic and homogeneous universe. Minkowski spacetime models the absence of gravity (or a negligible field). Similarly, different Newtonian spacetimes correspond to different physical situations. You'll have a spacetime for an uniform field (if you want an approximation for gravity near the surface of Earth) and you'll have a different spacetime for the detailed field of the Earth, and a different spacetime for the gravitational field of the Sun and all the planets in the solar system.
Why isn't Galilean spacetime equal to Newtonian spacetime? Geometrically, they are very different. When we talk about Galilean spacetime, we consider gravity to be a force. Schuller's notion of a Newtonian spacetime introduces gravity as spacetime curvature, so it is actually quite different from Galilean spacetime. I guess it is fair to say Galilean spacetime is a particular example of a Newtonian spacetime. Namely, it is the Newtonian spacetime that presents no gravitational effects (the connection is trivial).
