$\pi$ in arbitrary metric spaces
I believe I do have a partial answer to your question but I do not claim credit for it. http://www.jstor.org/stable/2687579 is the paper I am referring to. Undoubtedly Miha Habič is referring to the same thing. And here I will summarize the relevant info from the paper.
Only working with the $p$-norms defined in $\mathbb{R}^2$ as $$d_p((x_1,y_1),(x_2,y_2))=(|x_2-x_1|^p+|y_2-y_1|^p)^{1/p}$$ we already know that this is a norm if and only if $p\geq 1$ and the usual norms mentioned here like the taxicab, euclidean, and the max norm (by "setting" $p=\infty$) are all special cases so we only look at $d_p$ for $p\in [1,\infty)$.
The authors then derive the expression $$\pi_p=\frac{2}{p}\int_0^1 [u^{1-p}+(1-u)^{1-p}]^{1/p}du$$ for $\pi$ in any $p$-norm. Then they just numerically integrate and estimate $\pi$ for different $p$ and get
$$\begin{array}{ll} p & \pi_p \\ 1 & 4 \\ 1.1 & 3.757... \\ 1.2 & 3.572... \\ 1.5 & 3.259... \\ 1.8 & 3.155... \\ 2 & 3.141...=\pi \\ 2.25 & 3.155... \\ 3 & 3.259... \\ 6 & 3.572... \\ 11 & 3.757... \\ \infty & 4 \end{array} $$
Then the authors prove that the (global) minimum value of $\pi_p$ indeed occurs when $p=2$. And numerics seem to suggest that $\pi_p$ is always between $[\pi,4]$ so the answer to your question seems to be that there is no $p$-norm in which $\pi_p=42$.
One setting where this can be worked out is on 2-dimensional Riemannian manifolds (which have a metric in the metric space sense that's induced by the Riemannian metric). If you haven't heard about this before, Wiki has an introduction to what this means, with lots of pictures.
In that setting, you can pick a point $p \in M$, and there's a reasonable notion of a circle of radius $r$ centered at $p$, which I think agrees with yours. Then ask what is the circumference of a circle $L_r$ of radius $r$ centered at $p$. The corresponding value of $\pi$ would be $\pi_r = L_r/2\pi$. If you do a bit of work, you can work out that for small $r$, $\pi_r \approx \frac{L_r}{2 r} \approx \pi - \frac{\pi K}{6} r^2$, where $K$ is the sectional curvature of the surface at $p$. (This is exercise 5.7 in do Carmo's book on Riemannian Geometry)
So as $r$ gets small you get closer and closer to the usual value of $\pi$. Maybe the more interesting way to think about this is that the value of your $\pi$ tells you something about curvature: a space has positive curvature if small circles have circumference less $\pi r$, and negative curvature they have circumference more than $\pi r$.
I think your definition is reasonable and might produce some interesting results.
For example, if we put the taxicab metric on the plane, we should get $\pi_X$ = 4. Also, people say that if $\pi$ were 3, circles would be hexagons, which suggests that there's a metric on the plane that would make both of those true. I'd be curious to know what other $\pi_X$ values you could get.
You'll probably want some more conditions on your metric space. For example, probably the symmetries should act transitively on $X$ so that it doesn't matter where you put your disk. And maybe there should be scaling maps which fix a point but multiply all distances by a constant factor. In any case, to get a consistent $\pi_X$ you should be able to take a disk of any size.
When you finish doing these yourself, you can look up some literature by searching for "girth" of normed spaces.
A similar question was asked before on StackExchange Developing the unit circle in geometries with different metrics: beyond taxi cabs They came up with the ``elevator metric" and "bus "metric".
Any normed space should have notions of length, area and volume. So you can compute pi as
- the ratio of circumference to diameter
- the area of a unit circle
- half the circumference of unit circle
- 3/4 the volume of the unit circle (or whatever the appropriate fractions are in your normed space!)
Here is a paper called Volumes on Normed and Finsler spaces which looks helpful. There is also something called the isoperimetric problem, which describes a "circle" as maximizing the area given a length.