It's true that in many cases, for a (measurable) subset $S \subseteq \mathbb{R}^n$, you will be able to inherit a probability measure on $S$ from Lebesgue measure on $\mathbb{R}^n$, by defining $\mu(A) := \frac{m(A)}{m(S)}$ for $A \subseteq S$. It's also true that in this case, the Lebesgue measure of the Sierpinski triangle $S \subseteq \mathbb{R}^2$ is 0, meaning that definition won't work. So, we will have to go another route to construct a measure on $S$.
An analogous situation is that we can define a "uniform" measure on the unit sphere $S^n \subseteq \mathbb{R}^{n+1}$, even though $S^n$ has Lebesgue measure zero. (In this particular case, one way to construct the measure is to restrict the Riemannian manifold structure on $\mathbb{R}^{n+1}$ to $S^n$, then construct the volume form on $S^n$ corresponding to this Riemannian manifold structure on $S^n$. However, the Sierpinski triangle is certainly not a submanifold of $\mathbb{R}^2$, so that construction won't work here.)
One way to construct such a probability measure on $S$, which will be uniform, would be: start with the algebra of finite unions of subtriangles, and to each such object associate a measure in the natural way such that a subtriangle of level $n$ gets measure $3^{-n}$. Then, the construction of taking the corresponding outer measure, then forming the $\sigma$-algebra of sets measurable by the Caratheodory criterion will form a measure on $S$. (Though verifying that the initial measure satisfies the countable additivity condition will probably be fairly tricky, comparable to the trickiness of the check involved in constructing Lebesgue measure. This will be necessary to conclude that each subtriangle is measurable with respect to the final measure and has the expected measure $3^{-n}$.)
This measure will then, as expected, satisfy uniformity conditions such as translation invariance (when you take a translation from one subtriangle to another); and respecting contraction of $S$ to a subtriangle, with the contraction of $S$ to a subtriangle of level $n$ multiplying measures by $3^{-n}$. This measure will also certainly be singular with respect to Lebesgue measure (if you extend it to a measure on $\mathbb{R}^2$).