Do higher dimensions have axes?

They sure do. In three dimensions,

• the $x$-axis is the set of all points where both coordinates besides $x$ are zero,
• the $y$-axis is the set of all points where both coordinates besides $y$ are zero, and
• the $z$-axis is the set of all points where both coordinates besides $z$ are zero.

If you're looking at a $26$-dimensional space with coordinates $(a, b, c, \ldots, z)$, then

• the $a$-axis is the set of all points where all coordinates besides $a$ are zero, and
• the $b$-axis is the set of all points where all coordinates besides $b$ are zero, and
• the $c$-axis is the set of all points where all coordinates besides $c$ are zero, and
• ...
• the $z$-axis is the set of all points where all coordinates besides $z$ are zero.

Sure. In $n$ dimensions, the $k^{\text{th}}$ axis, for some $1 \leq k \leq n$, is the span of the vector whose $k^{\text{th}}$ component is $1$ and all other components $0$. The problem is that there's not a particularly satisfying geometric visualization of these axes and one largely has to settle for this algebraic abstraction.

The first important term here is that of a vector space. This is a general term; vector spaces can be far more than simply some $\mathbb{R}^n$. Polynomials, for example, form a vector space of infinite dimension.

Now you need to understand what a dimension is. If you have a vector in three dimensions, you write that vector like $(4,1,7)$. Those coordinates make sense in the context of a base. A base is a set of linearly independent vectors, which has the maximal possible size for that vector space. This size is the dimension. A three dimensional space is defined by having maximally three linear independent vectors in a set.

Edit: On the recommendation of Kaj Hansen, let me at this point make clear that a base of a given vector space has always the same amount of vectors. This is important, because this defines the dimension of the vector space.

The coordinates are now in relation to your base. If your base is $\{b_1, b_2, b_3\}$, the vector $(4,1,7)$ is the vector you get by the linear combination $4b_1 + 1b_2 + 7b_3$.

Now realize that this base can change. You can take three linear independent vectors different from your current base and make them the new base, resulting in all vectors getting different coordinates.

To give you an example when this is used, consider a 3D image in computer graphics that depicts one of those wooden puppets that artists use. There is a global coordinate system for the whole image, and the puppet has coordinates within that. But the limps have local coordinate systems which are in relation to the joints of the puppet. Hope this helps you.

Now, for the axes, an axis is simply the line created when you follow the direction of a base vector. Nothing more. They are arbitrary, and they do not even have to be orthogonal to each other. They are not a fundamental principle of a space. Coordinates are just numbers how we model it.

Another point of view, let's say that you are to determine the axes of reality, of the three-dimensional space we live in. In which direction would you point the first axis? Arbitrary. You decide on some direction for the first an then for the others, and then you write things in relation to those (or, better to say, the base vectors, as you also need a length). There is not the one objective way to create your axes.

Edit: Inserted from the comment of Kaj Hansen, note that usually one uses the standard base $\{e_k\}^n_{k=1}$ for $\mathbb{R}^n$, which consists of vectors $k_i$ that are all zero except for the $k^{th}$ component which is one. With those, a coordinate evaluates to a vector with the same numbers. Now, I brought this here and not above for a reason, and that is because we just talked about the "axes of reality", which do not exist. In the real world, there is no objective coordinate (1, 0, 0). For that reason, even the standard base is not "objective" or anything, but bound to some definitions we made. When we compute things in math, (1, 0, 0) is an objective thing, though, because in math, we live only in definitions and axioms.

With more dimensions, it is not different. You have a base and you work with it, and you can totally change the base.

Things might get difficult when you talk about physical spaces, though. Time behaves different as a dimension, and we are now in a Minkowski space, and things are getting complicated.

If you are interested in examples for vector spaces which aren't R^n, like the polynomial vector spaces or that of other functions, I can explain that too, but as this is long enough already, I'll stop for now.