You ask: "Is Wikipedia's definition of fractal the standard?" and right near the top of Wikipedia's page of fractals, we see the following definition:
A fractal is a mathematical set that has a fractal dimension that
usually exceeds its topological dimension and may fall between the
The statement that the fractal dimension may "fall between the integers" really adds nothing but, other than that, I would say that this is fairly standard; it is unquestionably the definition that was put forward by Mandelbrot around 1975 when he coined the term "fractal". He did not refer to "fractal dimension" at that time but, rather, the "Hausdorff-Besicovitch dimension" as he put it. In fairness, the usefulness of this definition has been debated with even Mandelbrot himself feeling that it might not be inclusive enough. Nonetheless, this comparison of dimension is central in fractal geometry. Gerald Edgar calls his great book, Measure, Topology, and Fractal Geometry, a meditation on the definition.
Taking this to be the definition, we can definitely say that the Cantor set satisfies it. If by "fractal dimension" you mean similarity dimension, then the Cantor set has fractal dimension $\log(2)/\log(3)$, since it's composed of two copies of itself scaled by the factor three. Also, the set is regular enough that any reasonable definition of fractal dimension agrees with that computation. (Well, any real-valued defintion.)
Topological dimension is a trickier thing, actually. It's inductive in nature. Totally disconnected sets (like single points, finite sets, or notably the Cantor set) have dimension zero. Higher dimensions are defined in terms of lower dimensions. The space we live in is three dimensions because balls in this space have a surface that is two dimensional. Because of this inductive nature, topological dimension always yields an integer.
When you write that you "do not see the irregular aspects or the complexity that is usually inherent with fractals", I think you might have a bit of a mis-understanding about fractal geometry. The Cantor set is indeed regular but, then so are all the strictly self-similar sets studied in classical fractal geometry - the Koch curve, the Sierpinski triangle, the Menger sponge, and countless others all display this regularity. Indeed, it's exactly this regularity that allows us to understand them.
To emphasize this regularity, and how it appears in not just the Cantor set, compare the following zooms of
The Cantor set
The Koch curve
Now, of course, there are "irregular" fractals - or, at least, less regular fractals. Examples include random version of self-similar sets, examples that arise from number theory, and examples arising from complex dynamics (like Julia sets). It's not their irregularity that makes these objects fractal, however. On the contrary, its the regularity that we can find that allows us to analyse these objects to the point where we can characterize them as fractal. Of course, this analysis is bit harder with these less regular examples.