Why is the median of an even number of samples the arithmetic mean? When calculating the median for an even number of samples, we sort the samples and take the arithmetic mean of the two in the middle.
This seems quite arbitrary - it assumes the best way to interpolate between the two samples is a linear interpolation.
Why not use the geometric or some other mean? Wikipedia lists quite a lot of options:

I guess one argument is that a choice needs to be made and any choice would add some bias. Is there a more fundamental reason?

When this happens, I see two options.

*

*Report the median as an interval, which gives us infinitely-many medians instead of just one. (This is not even a confidence interval or credible interval for a point estimate.)


*Figure out a way to pick one value.
Picking the arithmetic mean of the middle two points is just one way of doing the estimation. Advantages include the ease with which is is computed and explained. Disadvantages could include bias when there is a skewed distribution.
There are always many ways of estimating a quantity. In fact, the quantile function in R software has at least nine options for computing quantiles like the median. If you have reason to believe that an arithmetic mean of the middle two values has inferior properties to some other form of estimation, you are free to argue why your alternative is better. Plenty of people have seen other estimators that are not optimal for them and have proposed alternatives (e.g., James-Stein estimator).
EDIT
The following simulation shows the claim by John Madden in the comments that all values in the interval minimize the absolute deviation function whose minimizer is one way to define median.
set.seed(2023)
y <- c(1, 2, 3, 7, 8, 9)
candidate_medians <- seq(2, 8, 0.01)
median_loss <- rep(NA, length(candidate_medians))
for (i in 1:length(candidate_medians)){
  
  median_loss[i] <- mean(abs(y - candidate_medians[i]))
  
}
plot(candidate_medians, median_loss)


All values $M$ between the middle data values, $3$ and $7$ (so $M\in[3,7]$), give equal and minimizing values of $L(y, M) = \frac{1}{n}\sum_{i=1}^n\big\vert y_i-M\big\vert$.
