A bit of setup.
One way of talking about whether or not two things are "the same" is to ask if they are isomorphic objects in some category. Broadly speaking, two objects are isomorphic if there is a bijection between them which preserves whatever structure is interesting about those objects.
Fractals are, perhaps, best understood as metric spaces. The natural morphisms between metric spaces are isometries, which are functions which preserve distances. Reflections, translations, and rotations are all isometries. Scalings are not isometries, but it is certainly reasonable to consider two fractals to be the same if one is a scaled copy of the other, so perhaps the right morphisms are isometries with scalings. To make this formal, I'll introduce some definitions:
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. An isometry between $X$ and $Y$ is a bijective map $\varphi : X \to Y$ such that for all $x_1, x_2 \in X$,
$$ d_Y(\varphi(x_1), \varphi(x_2)) = d_X(x_1, x_2). $$
A isometric scaling[1] is a bijective map $\psi : X \to Y$ such that there is some $C > 0$ such that for all $x_1, x_2 \in X$,
$$ d_Y(\varphi(x_1), \varphi(x_2)) = C d_X(x_1, x_2). $$
$C$ is called the scaling ratio of the isometric scaling. Note that the inverse of an isometry is an isometry, and that the inverse of a scaling isometry is a scaling isometry.
While the term "fractal" is not well-defined (there is no universally agreed upon notion in the mathematical community of what a fractal is), we can talk more broadly about metric spaces. For the purposes of this discussion, two metric spaces are "the same" if there is an isometric scaling between them. That is, we are working in a category where the objects are metric spaces, and the morphisms are scaling isometries.[2]
A very common question in this kind of setting is "Are there any ways of distinguishing objects in a category without actually checking to make sure that they are not isomorphic?" For example, in topology, if two spaces have different fundamental groups, then they are not homeomorphic. The fundamental group is thus an invariant in the category of topological spaces. Similarly, the Hausdorff dimension is an invariant in the category of metric spaces with scaling isometries---if two spaces are scaling isometric to each other, then they must have the same Hausdorff dimension. The original question is, then, how strong an invariant is the Hausdorff dimension?
A question and answer.
Question: Suppose that $(X,d_X)$ and $(Y,d_Y)$ are metric spaces. Further suppose that
$$ \operatorname{dim}_{H}(X) = \operatorname{dim}_H(Y), $$
where $\operatorname{dim}_H(X)$ denotes the Hausdorff dimension of $(X,d_X)$. Must it be true that $(X,d_X)$ and $(Y,d_Y)$ are scaling isometric?
The answer to the question is "no"; two spaces may have the same Hausdorff dimension and, nevertheless, be distinct (i.e. not scaling isometric).
Examples.
A simple example. The simplest example I can think of is to take $X = \{a\}$ (i.e. a singleton set) and $Y = \{\alpha, \beta\}$, each with the trivial metric ($d(x,y)= 1$ if $x\ne y$). These two spaces both have Hausdorff dimension zero, but there is no scaling isometry between them (a scaling isometry is, by definition, bijective, and there cannot be a bijective map between a singleton set and a set with two elements.
Intervals. If $a < b$, then the open interval $(a,b)$ and the closed interval $[a,b]$ each have Hausdorff dimension $1$. However, these two spaces are not scaling isometric: because one interval is open and the other is closed, you cannot find a scaling isometry between them.
Self-similar sets. Let $\{\varphi_j : X \to X \}_{j=1}^{N}$ be a collection of scaling isometries, where $\varphi_j$ has scaling ratio $C_j < 1$. Such a collection of maps is called an iterated function system. Define the map $\Phi : \mathscr{P}(X) \to \mathscr{P}(X)$ defined by
$$ \Phi(E) = \bigcup_{j=1}^{N} \varphi_j(E). $$
There exists a unique nonempty compact set $\mathscr{A}$ such that
$$ \mathscr{A} = \Phi(\mathscr{A}), $$
called the attractor of $\Phi$. Under relatively mild conditions, the Hausdorff dimension of $\mathscr{A}$ is the unique real solution $s$ to the Moran equation
$$ \sum_{j=1}^{n} C_j^s = 1. $$
For example, the usual ternary Cantor set is the attractor of the system $\{\varphi_1, \varphi_2\}$ given by
$$ \varphi_1(x) = \frac{1}{3}x, \qquad \varphi_2(x) = \frac{1}{3} x + \frac{2}{3}. $$
Both maps have contraction ration $1/3$, and so the Hausdorff dimension of the Cantor set is $s$, where
$$ 1 = \sum_{j=1}^{2} C_j^s = 2\left( \frac{1}{3} \right)^s
\implies \left( \frac{1}{3} \right)^s = \frac{1}{2}
\implies s \log(3) = \log(2)
\implies s = \frac{\log(2)}{\log(3)} = \log_3(2).
$$
The von Koch curve[3] can be realized as the attractor of an iterated function system consisting of four maps, each with ratio $1/3$. There is a nice writeup of the procedure on Larry Riddle's website.
The two-dimensional Cantor dust can also be realized as the attractor of an iterated function system consisting of four maps, each with ratio $1/3$.
Both of these spaces, with the induced metric from $\mathbb{R}^2$, have Hausdorff dimension $\log_3(4)$. However, they are not scaling isometric. The most straight-forward argument is topological: the von Koch curve is connected, while the Cantor dust is totally disconnected. Scaling isometries are homeomorphisms, and homeomorphism preserve connectedness, hence no scaling isometry can be found between these spaces.
Some further discussion.
In a comment Dave L. Renfro suggests that the dimension of a set is something like the "weight" or "mass" of that set. I think that this is a good start at an analogy for the meaning of dimension, but (for me) it is not quite right.
In mathematics, a measure is a tool which is used determining some generalized notion of mass—the measure of a set is analogous to the mass of that set. If we assume that sets come from spaces of uniform density, then a one-dimensional set's mass is proportional to its length, while a two-dimensional set's mass is proportional to its area. Unfortunately for the analogy, the dimension of a set doesn't say much about the measure (mass or weight) of that set. For example, the interval $[0,1]$ and the real line are both one-dimensional, but the interval has finite one-dimensional measure (i.e. finite mass), while the real line has infinite one-dimensional measure.
Rather, the dimension indicates which tool is appropriate for determining the measure of that set. For example, a set with Hausdorff dimension $1$ is naturally measured in terms of a linear measure like inches, kilometers, or light-years, while a set with Hausdorff dimension $2$ is naturally measured in terms of areas, such as square meters or acres. The fact that both the real line and the interval $[0,1]$ are one-dimensional tells us that both can be measured with the same tool (length), even though their actual measures are quite different.
In short, the dimension of a space tells us which tool to use if we want to measure the space.
[1] I'll note that the term "isometric scaling" is not one that I know to be standard—I just made it up. A quick Googling shows that this term does appear, but it seems to be mostly related to descriptions of biological systems. If you used the term "isometric scaling" in a mathematical context, it is likely that you would be understood, but it is also possible that, without clarification, you would cause confusion.
[2] There is actually some work which needs to be done in order to show that this really is a category, but it should all work out in the end. Also, this is probably a bit more abstract than is really necessary to ask-and-answer this question.
[3] Often, the von Koch curve refers to the object created by replacing each side of an equilateral triangle of side length $1$ with four segments of length $1/3$. In this discussion, I mean the curve obtained by applying this same procedure to a single segment.
