tl;dr Size plays a role in this very specific scenario in kind of an indirect way, but generally speaking, variability in its enormous scope depends on a lot of other things more for other kinds of objects, and this is not a relation to be applied outside of maybe pulsars (probably)
Let me know if some of these assumptions I’m making here are wrong, but I’m going to make some so that I can answer the question as I understand you’re asking it.
When you say ‘oscillations’, I’m going to assume that you’re speaking of oscillations regarding the amplitude of the photons, more commonly referred to as the intensity or flux of the object.
For context, the relation between flux and intensity is that intensity is the strength of the light at the surface of the object, and the flux is the measured intensity at a given distance away (which may or may not be known). From here on out I will just use flux; since we’re dealing with relative changes and flux is what we measure, it makes the most sense here.
The attribute of oscillation of the flux over time is called variability, and celestial objects that have this behavior are referred to as variables, variable stars, variable objects, etc.
There are an immense amount of different types of variable with a wide range of reasons and attributes. Almost any kind of object we see in the night sky can be a variable; stars, galaxies, neutron stars, supernovae, etc.
Some objects are periodic variables, meaning they repeat there oscillations for an extended amount of time. They may evolve and change or cease variability, but they’re periodic for a time (examples: delta Scutti stars, the North Star, most red giants, etc.)
Other objects are variable but for just a short amount of time; for example, cataclysmic variables are a one-and-done kind of change in light, like supernovae (cataclysmic), or a helium flash (non-cataclysmic). These are non-periodic.
Since you specifically mentioned oscillations, I will specifically talk about periodic variables.
To say there’s a lot of reasons you can have a periodic variable would be an understatement, so let’s consider very specifically the example mentioned in the paper that prompted your question, and extrapolate from there.
What is being heavily implied by the article is that ‘the cow’ is a neutron star; an extremely compact object that is just short of total gravitational collapse into a black hole. For the most part, neutron stars come from the remnants of dying stars, and are situations where gravity is so intense that matter is on its final limb and is being sustained only by neutron degeneracy pressure, or in other words, neutrons really don’t want to be in the same state so they resist the gravitational collapse that keeps them held together.
With that background, there are two main principles that allow neutron stars to be variables; conservation of angular momentum, and extremely intense magnetic fields.
For the magnetic fields, the important takeaway is that because of the shape of the magnetic field, neutron stars can eject highly energized light and charged particles from its poles, and these poles can be facing towards us or away from us, and we see more light naturally when they face us than when they don’t.
The conservation of angular momentum states that if you have a spinning, extended object and you have no external torques, and you shrink the object, it’s going to spin faster. This can be felt in a spinning chair or watching an ice skater; as they or you pull your arms in while spinning, you spin faster. This essentially happens at an extreme scale with a star; you had all this matter that was pretty voluminous starting out (think of the size of the sun, these objects are over one solar mass) and you shrink it down so that you could draw it within the United States. Conservation of angular momentum means that this small object is going to move very, very fast.
So we come back to your original question, but now with context. These neutron stars act almost like lighthouses spinning at insane speeds. We thus get the 4.4 millisecond period in its variability, and we see why in this case size matters; because neutron stars have shrunk from previously massive objects, their small size makes them spin faster, making more variability. Because of their pulsing behavior, these neutron stars are more specifically labeled pulsars.
This principle of smaller sizes leading to faster variables does not extend to almost any variables, and even is kind of a strange way to describe pulsars in relation to each other. For a first time explanation, yes, their small size is the principal reason for why they have such small periods, but all pulsars are small, and the speed of given pulsar can also depend on the nature of its creation and the conditions of its former stellar life.
Other variables vary because of a wide variety of reasons. From internal structure, to transiting exoplanets, to binary systems, there is a massive amount of reasons, a lot of which are not size dependent (or if they are, size is often not the main factor).