How well does breaking multiple DH key exchanges over the same group scale?
The simplest index-calculus attack on discrete logarithms is the following. You have a generator $g$, a target $y$ and a bunch of small primes $\ell_1, \dots, \ell_k$. The computation proceeds in three phases.

First generate lots of relations of the form
$$g^{r_i} = \prod_j \ell_j^{s_{ij}}.$$

These relations give you a set of linear equations in $r_i$, $s_{ij}$ and unknown $\log_g \ell_j$. Solve this system for the unknown $log_g \ell_j$.

Now find one relation of the form
$$y g^u = \prod_j \ell_j^{v_j}.$$
From this, the logarithm of $y$ is easily recovered.

The third phase is much, much cheaper than the two first phases. Notice also that the two first phases do not involve $y$, so when you want to compute the logarithms of multiple elements, you only have to repeat the third phase.

I believe these essential characteristics are shared with the more advanced d.log.algorithms, so the cost of computing $n$ logarithms is not $n$ times the cost of computing one logarithm, but merely somewhat larger than computing one logarithm.

Typically, our goal is to make it impossible to compute even one logarithm, so this does not have to be a reason for using different groups. There are good reasons for everyone to use the same group. For prime finite fields, you can use carefully chosen primes that make arithmetic fast and/or secure implementation easy. For elliptic curves, you can use carefully chosen primes and curves that make arithmetic fast and/or secure implementation easy. (Note that index calculus doesn't work for elliptic curves, but I think there are speedups for Pollard $\rho$ when computing multiple discrete logarithms.)