If all motion is relative, how does light have a finite speed?

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.

First: Maxwell's equations predict that the speed of light is absolute. The whole motivation for the special theory of relativity is to reconcile this with the notion that all motion is relative. In other words, you're worried about exactly the same thing that troubled Einstein. You just haven't understood how he solved it.

The key to your confusion is right here:

If one object is moving (uniformly) at 60% c and another object is also moving at 60% c but in the exact opposite direct, then from the perspective of either one (if they could still see each other) the other would appear to violate that speed limit.

This isn't true. In fact, from the perspective of either one, the other is moving away at a speed of about $.88c$. Of course you, standing on the ground, will claim that they are moving away from each other at $1.2c$. This is possible because observers in motion relative to each other will disagree about things like the distance between two events and the time between those events --- and therefore will disagree about the speeds at which things are moving away from each other. Special relativity tells you exactly how to calculate those disagreements.

By my reckoning, if all speed is relative, then no mater how fast you go light should always race away from you at the same apparent speed. I.e. there should be no speed limit.

If an invariant speed $c$ exists, then if an entity has speed $c$ relative to an inertial reference frame (IRF), the entity has speed $c$ relative to all IRFs.

That's what it means for there to be an invariant speed.

Now, if you think about that for a little bit, it follows that an entity with speed less than (or greater than) $c$ in an IRF, cannot have speed $c$ in any IRF.

Thus $c$ is a limiting speed in this sense: entities either have speed $c$ in all IRFs or an IRF exists in which the entity has a speed that is arbitrarily close to $c$.

For further reading, I recommend this paper, Nothing but Relativity, in which it is shown that, assuming only the principle of relativity, the most general coordinate transformation involves an invariant speed.

We deduce the most general space-time transformation laws consistent with the principle of relativity. Thus, our result contains the results of both Galilean and Einsteinian relativity. The velocity addition law comes as a bi-product of this analysis. We also argue why Galilean and Einsteinian versions are the only possible embodiments of the principle of relativity.