Q. What are all the numbers $n$, such that each is represented as repeated digits—different digits—in two different bases $b_1$ and $b_2$.
So in base $b_1$, $$ n_{b_1} = c c c \cdots c \;, $$ and in base $b_2$, $$ n_{b_2} = d d d \cdots d \;, $$ and $c$ and $d$ are digits with $c \neq d$.
Such numbers exist: for example, $$ 777_{10} = 3333_{6} \;. $$ I would be interested in a characterization of such $\{n, b_1, b_2 \}$.
An obvious extension is: repeated digits in $k$ bases $b_1, b_2, \ldots, b_k$. Added. It would be especially nice to find a number such that: $$ 1 1 \cdots 1 _{b_1} = 2 2 \cdots 2 _{b_2} = 3 3 \cdots 3 _{b_3} = \cdots $$
In base $b$ the numbers that have digit $d < b$ repeated $n$ times can be described as:
$$d \cdot\frac{b^n - 1}{b - 1}$$
Thus your question can be simplified to finding integer solutions for:
$$d \cdot\frac{b_1^n - 1}{b_1 - 1} = c \cdot\frac{b_2^m - 1}{b_2 - 1}$$ $$d \cdot (b_2 - 1) \cdot(b_1^n - 1) = c \cdot (b_1 - 1) \cdot (b_2^m - 1)$$
With $d < b_1$ and $c < b_2$.