Plane partitions with equal margins
I don't know if this is optimal, but here is a pair of plane partitions of $n=53$ with no symmetries that have equal projections on each individual axis:
$$A=\begin{matrix} 5 4 4 4 4 \\ 5 4 1 1 \ \ \\ 5 2 1 1 \ \ \\ 5 1 1 1 \ \ \\ 1 1 1 1\ \ \end{matrix}$$ and $$B=\begin{matrix} 5 5 5 5 1 \\ 4 4 1 1 1 \\ 4 2 1 1 1 \\ 4 1 1 1 1 \\ 4 \ \ \ \ \ \ \ \ \end{matrix}$$ This is, of course, a bit like cheating because we just overlapped two cyclically symmetric partitions that are transpose with an L shape to break any symmetries.
If the question doesn't care about symmetries, and is simply asking for the smallest $n$ for which some plane partition of $n$ is not uniquely determined by the projection triple $(\lambda,\mu,\nu)$ then John Machacek's example is the minimal one.
Proposition: For $n\le 12$ a plane partition of $n$ is uniquely determined by the triple $(\lambda, \mu, \nu)$.
Proof: It is simple to check that for a classical partition $\alpha$, if $\alpha$ is not determined by the length of its longest row and longest column, $(\alpha_1, \alpha'_1)$, then we must have $\min(\alpha_1,\alpha'_1)\geq 3$ and $|\alpha|-\alpha_1-\alpha'_1\geq 1$.
Now let's denote the rows of our plane partition as $\beta^1, \dots, \beta^r$ where each $\beta^i$ is a partition. If $\beta^1$ doesn't satisfy the conditions of the previous paragraph then we could reconstruct $\beta^1$ from the knowledge of its longest row and column. Both of these can be found from the information of the projections of the plane partition. Once we reconstruct $\beta^1$ we can repeat the process and reduce the problem to reconstructing $\beta^2, \dots, \beta^r$. Eventually we recover the whole partition. Moreover we can read the plane partition using a different coordinate plane and obtain the same information for $\gamma^1$ and $\delta^1$ which denote the slices of our partition on the planes $y=0$ and $z=0$.
Since each of $\beta^1, \gamma^1, \delta^1$ satisfy the conditions of the first paragraph we have that the size of our plane partition is at least $$|\beta^1|+|\gamma^1|+|\delta^1|-\frac{1}{2}((\beta^1)_1+(\beta^1)'_1+(\gamma^1)_1+(\gamma^1)'_1+(\delta^1)_1+(\delta^1)'_1)+1$$ $$\geq |\beta^1|+|\gamma^1|+|\delta^1|-((\beta^1)_1+(\beta^1)'_1+(\gamma^1)_1+(\gamma^1)'_1+(\delta^1)_1+(\delta^1)'_1)+10\geq 13$$ as desired.
Hopefully I understood the definitions and computed correctly. How about $n = 13$ with $$A = \begin{matrix} 3 3 1 \\ 2 1 1 \\ 2 \ \ \end{matrix}$$ and $$B = \begin{matrix} 3 2 2 \\ 3 1 \ \\ 1 1 \ \end{matrix}$$ where all margins are $(7,4,2)$.
Here is a smaller example than Gjergji Zaimi's that has at least less symmetry than John Machacek's. Among the plane partitions of 14, $$ A = \; \begin{matrix} 3 & 2 & 2 & 1 \\ 3 & 1 \\ 1 & 1 \end{matrix} $$ and $$ B = \; \begin{matrix} 3 & 3 & 1 & 1 \\ 2 & 1 & 1 \\ 2 \end{matrix} $$ both have margins $((8,4,2), (7,4,2,1), (8,4,2))$.
The smallest pairs with equal distinct margins occur with $n = 16$, e.g., $$ C = \; \begin{matrix} 5 & 2 & 2 & 1 \\ 3 & 1 \\ 1 & 1 \end{matrix} $$ and $$ D = \; \begin{matrix} 5 & 3 & 1 & 1 \\ 2 & 1 & 1 \\ 2 \end{matrix} $$ both have margins $((10, 4, 2), (9, 4, 2, 1), (8, 4, 2, 1, 1))$.
(I was able to get Mathematica to check for equal margins among all plane partitions through $n = 16$ and might be able to go a little farther. Luckily these examples came up within the bounds of my programming ability and computer power.)