Is the age of the universe relative to an observer's location in that universe? According to Wiki the age of the universe is 13 billion years old, and I was taught that background radiation made the universe uniform in all directions.

Doesn't this define a sphere of space in the universe with Earth at the center? This sphere would have a radius of 13 billion light years.

We (on Earth) could not see anything outside this sphere. As light did not exist more than 13 billion years ago. So for us this defines some kind of edge for the universe.

But what happens when an alien in another galaxy 5 billion light years away from Earth also looks towards the stars. Don't they see a uniform background radiation and measure the same age of the universe?

That alien would also have a sphere of 13 billion years.

So we have two spheres, one for Earth and one for our alien friend.

These two spheres would overlap each other by 5 billion years. Which means, if we look in the opposite direction from that alien's galaxy. We can see an additional 5 billion years further than they can.

If we measure the total width of the two overlapping spheres we get a distance of 18 billion light years.

I now grasp my hair in confusion. Is this possible that two visible spheres could overlap to create a distance longer than the oldest ray of light? And from there I fall into more questions. Can two people measure the same universe age from different spots in the universe.

That's why I think my idea is is wrong, but how could it be wrong and why?
You are labouring under the misapprehension that how far we can see directly gives the age of the universe. Whilst it is true that the oldest light we can see was emitted some 13.7 billion years ago, the stuff that emitted that light is now roughly 46 billion light years away, thanks to expansion of the universe.

The universe itself probably extends significantly beyond this and may be infinite. That does not conflict with the big bang model or relativity. If the universe is infinite then it has always been infinite. Places that are separated by more than the oldest light ray as you put it, are not in causal contact and, depending on the cosmological parameters, may never be.

In a homogeneous, isotropic universe (and don't forget we can only make assumptions about regions beyond our observable universe), all observers would agree on the age of the universe and that the universe was once very much smaller and all would see (on average) the same sort of universe in all directions.

To ensure that places that are outside of causal contact now are homogeneous requires that they were in causal contact in the past. This is the nature of the "horizon problem"".

I took the picture below from the "slideplayer" website, I am unsure who the originator is, but it serves its purpose here and I think encapsulates your question. It appears in this diagram that A and B can never have "communicated" and can never have originated from the same place. The solution provided by inflation is to have a massive exponential expansion of space in the first fraction of a second. In short, the distances between points in space (or galaxies if you like, although there were no galaxies at the epoch of inflation) are suddenly increase by many, many orders of magnitude. This give the appearance of faster-than-light motion, although the cosmic speed limit only applies to local measurements and not to the expansion of space itself. The end result is a universe that appears homogeneous well beyond the limits of a radius of the age of the universe in light years.