It sounds like your confusion is coming from taking paraphrasing such as "everything is relative" too literally. Furthermore, this isn't really accurate. So let me try presenting this a different way:
Nature doesn't care how we label points in space-time. Coordinates do not automatically have some real "physical" meaning. Let's instead focus on what doesn't depend on coordinate systems: these are geometric facts or invariants. For instance, our space-time is 4 dimensional. There are also things we can calculate, like the invariant length of a path in space-time, or angles between vectors. It turns out our spacetime has a Lorentzian signature: roughly meaning that one of the dimensions acts differently than the others when calculating the geometric distance. So there is not complete freedom to make "everything" relative. Some relations are a property of the geometry itself, and are independent of coordinate systems. I can't find the quote now, but I remember seeing once a quote where Einstein wished in reflection that instead of relativity it was the "theory of invariants" because those are what matter.
Now, it turns out that the Lorentzian signature imposes a structure on spacetime. In nice Cartesian inertial coordinates with natural units, the geometric length of a straight path between two points is:
$ds^2 = - dt^2 + dx^2 + dy^2 + dz^2$
Unlike space with a Euclidean signature, this separates pairs of points into three different groups:
$> 0$, space like separated
$< 0$, time like separated
$= 0$, "null" separation, or "light like"
No matter what coordinate system you choose, you cannot change these. They are not "relative". They are fixed by the geometry of spacetime. This separation (light cones if viewed as a comparison against a single reference point), is the causal structure of space time. It's what allows us to talk about event A causing B causing C, independently of a coordinate system.
Now, back to your original question, let me note that speed itself is a coordinate system dependent concept. If you had a bunch of identical rulers and clocks, you could even make a giant grid of rulers and put clocks at every intersection, to try to build up a "physical" version of a coordinate system with spatial differences being directly read off of rulers, and time differences being read from clocks. Even in this idealized situation we cannot yet measure the speed of light. Why? Because we still need to specify one more piece: how remote clocks are synchronized. It turns out the Einstein convention is to synchronize them using the speed of light as a constant. So in this sense, it is a choice ... a choice of coordinate system. There are many coordinate systems in which the speed of light is not constant, or even depends on the direction.
So, is that it? It's a definition?
That is not a very satisfying answer, and not a complete one. What makes relativity work is the amazing fact that this choice is even possible.
The modern statement of special relativity is usually something like: the laws of physics have Poincare symmetry (Lorentz symmetry + translations + rotations).
It is because of the symmetry of spacetime that we can make an infinite number of inertial coordinate systems that all agree on the speed of light. It is the structure of spacetime, its symmetry, that makes special relativity. Einstein discovered this the other way around, postulating that such a set of inertial frames were possible, and derived Lorentz transformations from them to deduce the symmetry of space-time.
So in conclusion:
"If all motion is relative, how does light have a finite speed?"
Not everything is relative in SR, and speed being a coordinate system dependent quantity can have any value you want with appropriate choice of coordinate system. If we design our coordinate system to describe space isotropically and homogenously and describe time uniformly to get our nice inertial reference frames, the causal structure of spacetime requires the speed of light to be isotropic and finite and the same constant in all of the inertial coordinate systems.
Backing up what zeldredge said, what you asked about is known as "relativity without light". According to the intro of this paper (arXiv link) for instance, the original argument was given as early as 1910 by Ignatowski, and has been rediscovered several times. There is a modern version due to David Mermin, in "Relativity without light", Am. J. Phys. 52, 119-124 (1984), but a pretty accessible presentation may also be found in Sec.2 of this paper by Shan Gao: "Relativity without light: A further suggestion" (academia.edu link). The basic idea is that the existence of an invariant speed follows directly from the homogeneity and isotropy of space and time, and the principle of relativity. No reference to a speed limit is needed, but it does follow that the invariant speed acts as a speed limit. The only alternative is a universe without a speed limit (infinite invariant speed), where kinematics is governed by the Galilei transformations. Why it is that our universe has a finite invariant speed, and not an infinite one, remains an open question. Gao's "further suggestion" is that the invariant speed is related to the discreteness of space and time at the Plank scale, which is an intriguing thought in its simplicity, but then it remains just a "thought" so far.
Best Answer
Imagine that there is a person who prefers to measure the amount of money in his bank account with the value $V$. The equation is $V = C\tanh N$, where $N$ is the actual amount of money in dollars. This person will also be confused:
The answer is that he is just using a "wrong" variable to measure his assets. $V$ is not additive — it is a transform of an additive variable, $N$, which he has to use in order for everything to make sense. And there is no "law of the universe" that limits the value of $V$ — such a limit is just a product of his own stubbornness.
The same thing applies to measure speed — it is the "wrong" variable to describe the rate of motion; speed is not additive. The "correct" variable is called "rapidity" — it is additive, and there is no limit on it.