The Schwarzschild metric describes the spacetime of a black hole. A common way to express a metric is through the corresponding line element, which tells us how to measure distance in the spacetime:

$$ds^2 = -(1+2\varphi)\,c^2dt^2 + (1+2\varphi)^{-1}dr^2 + r^2 d\Omega^2,$$

where $\varphi = -\frac{GM}{c^2\,r}$ is the ratio in the question. And $-\frac{GM}{r}$ is the Newtonian gravitational potential.

In the limit where $\varphi\rightarrow 0$ the Schwarzschild metric reduces to the Minkowski metric for flat spacetime (in spherical coordinates).

It is common to compute a series expansion of the Schwarzschild metric for small $\varphi$. This is a weak gravity limit, as the gravitational potential $\varphi$ acts like a small perturbation on flat, Minkowski space. In this limit the effects of the black hole's gravity reproduce the dynamics of Newtonian gravity to lowest order. Relativistic corrections to Newtonian gravity appear as terms of $\varphi^2$ or higher order.

If $\varphi\ll 1$, then the lowest order approximation, i.e. Newtonian gravity, is a good approximation to the dynamics. If $\varphi\sim 1$, then Newtonian gravity will not be a good fit to the dynamics.

Depending on how accurately you need to model the dynamics, relativistic corrections can be important for much smaller $\varphi$s than you might think! For the sun, $\frac{GM}{c^2} \approx 1.5$ km and Mercury's closest approach to the sun during its orbit is $r\approx 46\times 10^6$ km. Despite the very small $\varphi$ for Mercury's orbit, we measure its dynamics well enough to notice the difference between Newtonian gravity and GR.

The Schwarzschild metric describes the spacetime of a black hole. A common way to express a metric is through the corresponding line element, which tells us how to measure distance in the spacetime:

$$ds^2 = -(1+2\varphi)\,c^2dt^2 + (1+2\varphi)^{-1}dr^2 + r^2 d\Omega^2,$$

where $\varphi = -\frac{GM}{c^2\,r}$ is the ratio in the question. And $-\frac{GM}{r}$ is the Newtonian gravitational potential.

In the limit where $\varphi\rightarrow 0$ the Schwarzschild metric reduces to the Minkowski metric for flat spacetime (in spherical coordinates).

It is common to compute a series expansion of the Schwarzschild metric for small $\varphi$. This is a weak gravity limit, as the gravitational potential $\varphi$ acts like a small perturbation on flat, Minkowski space. In this limit the effects of the black hole's gravity reproduce the dynamics of Newtonian gravity to lowest order. Relativistic corrections to Newtonian gravity appear as terms of $\varphi^2$ or higher order.

If $\varphi\ll 1$, then the lowest order approximation, i.e. Newtonian gravity, is a good approximation to the dynamics. If $\varphi\sim 1$, then Newtonian gravity will not be a good fit to the dynamics.

Depending on how accurately you need to model the dynamics, relativistic corrections can be important for much smaller $\varphi$s than you might think! For the sun, $\frac{GM}{c^2} \approx 1.5$ km and Mercury's closest approach to the sun during its orbit is $r\approx 46\times 10^6$ km. Despite the very small $\varphi$ for Mercury's orbit, we measure its dynamics well enough to notice the difference between Newtonian gravity and GR.

To address the second point, the ratio isn't quite density. The density of an object is $\rho = M/r^3$, and the ratio is proportional to $M/r$. The Schwarzschild radius of a black hole is proportional to its mass $r_s = \frac{2GM}{c^2}$. If we take the size of a black hole to be $r_s$, then we can calculate the average density of everything inside the event horizon. For a $10$ solar mass black hole, similar in size to several LIGO has detected, $$\varphi = \frac{GM}{c^2\,r_s} = \frac{1}{2}, \quad\quad \rho = \frac{M}{{r_s}^3} \approx \frac{(10)(2\times 10^{30}\,\mathrm{kg})}{((10)(3\, \mathrm{km}))^3} = 7\times 10^{14} \, \mathrm{g/cm}^3. $$ The super massive black hole at the center of M87, that's in the Event Horizon Telescope's famous image, has a mass of about $6.5$ billion solar masses. For this black hole $$\varphi = \frac{GM}{c^2\,r_s} = \frac{1}{2}, \quad\quad \rho = \frac{M}{{r_s}^3} \approx \frac{(6.5\times 10^9)(2\times 10^{30}\,\mathrm{kg})}{((6.5\times 10^9)(3 \,\mathrm{km}))^3} = 0.0017 \, \mathrm{g/cm}^3. $$

Relativistic effects are equally important near the event horizon of both, but M87's black hole has an average density on the order of atmospheric air.