Does π depends on the norm? If we take the definition of π in the form:

π is the ratio of a circle's circumference to its diameter.

There implicitly assumed that the norm is Euclidian:
\begin{equation}
   \|\boldsymbol{x}\|_{2} := \sqrt{x_1^2 + \cdots + x_n^2}
\end{equation}
And if we take the Chebyshev norm:
\begin{equation}
   \|x\|_\infty=\max\{ |x_1|, \dots, |x_n| \}
\end{equation}
The circle would transform into this:

And the π would obviously change it value into $4$.
Does this lead to any changes? Maybe on other definitions of π or anything?

Under Euclidean metric there are number of constants that their values coincide and are collectively denoted by the symbol $\pi$. 
How ever some of the coinciding values are independent from the metric and some are coupled with the metric and the geometry under consideration. 
In your example, would the calculation of areas remain the same? How does the value of calculation under new metric need to be adjusted for unit square? Would the $\pi$ of calculation of area be the same one as the one for calculation of perimeter?
