We tested general relativity with effects as gravitational lensing and existence of black holes (Schwarzschild metric). But there are other metrics, e.g. Kerr-Newman metric for a point mass with charge and angular momentum.
Have we tested such metrics? Have we observed black holes with angular momentum at least?
For Kerr black holes, that is one of the tests performed by the Event Horizon Telescope (do notice the remark by Tim Rias on the comments: the EHT still doesn't have enough resolution to provide a significant test). Quoting the abstract from one of their papers (DOI: 10.3847/2041-8213/ab0f43),
Overall, the observed image is consistent with expectations for the shadow of a spinning Kerr black hole as predicted by general relativity. If the black hole spin and M87's large scale jet are aligned, then the black hole spin vector is pointed away from Earth. Models in our library of non-spinning black holes are inconsistent with the observations as they do not produce sufficiently powerful jets. At the same time, in those models that produce a sufficiently powerful jet, the latter is powered by extraction of black hole spin energy through mechanisms akin to the Blandford–Znajek process.
I should point out that, in astrophysical situations, the charge of a black hole is fairly negligible. I'll give the same argument given in Wald's General Relativity, p. 314. In geometrized Gaussian units ($G = c = 4\pi \epsilon_0 = 1$), the ratio between a proton's charge and its mass is $q/m \sim 10^{18}$. For an electron, $q/m \sim 10^{21}$. If you have a black hole with charge $Q$ and mass $M$, the ratio between electromagnetic and gravitational force it exerts on a particle of charge $q$ and mass $m$ is roughly $\frac{qQ}{mM}$. For a proton, we then have a ratio of roughly $10^{18} \frac{Q}{M}$, meaning the electromagnetic interaction is way more relevant. As a consequence, it is very difficult for a black hole to get a relevant amount of charge (say something beyond $Q/M \sim 10^{-18}$), because it will start to repel particles of same charge. Hence, in practice, effects due to charge are likely negligible in astrophysical situations.
The 'observation' of black holes is somewhat less direct than some other astrophysical bodies. What happens is that there is an accumulation of information, such as X ray emission, orbit parameters of other material (stars near the galactic centre, or orbiting dust in the case of some other candidates), and, more recently, gravitational waves, and one proposes that a black hole is the most likely body to account for the data. Essentially all black holes in practice are Kerr rather than Schwarzschild because it is very unlikely that the angular momentum would happen to be zero or even near zero. As regards "testing the metric", some observations are sensitive to some aspects of the metric, some are sensitive to others. The gravitational wave signals allow one to infer rough estimates of the angular momentum of the black holes involved, but not details of the metric. Observations of the binary pulsar PSR B1913+16 (Hulse-Taylor) do not concern a black hole but they concern neutron stars with angular momentum and constitute one of the most precise tests of strong-field general relativity. Observations by the "Event Horizon Telescope" are discussed in the answer here by Nickolas Alves.
