I think what you have established here is just that $rho$ tends to increase with mass. The density of planets isn't constant.
Let $rho = rho_0 (M/M_{earth})^{alpha}$, so that $M = (4/3)pi R^{3} rho_0 (M/M_{earth})^{alpha}$
Then
$$g = frac{GM}{R^2} = frac{4pi G}{3} R rho$$
Replace $R$ with $(3M/4pi rho)^{1/3}$ so that
$$ g = frac{4pi G}{3} left(frac{3M}{4pi rho}right)^{1/3} rho$$
$$ g = left(frac{4pi}{3}right)^{2/3} G M^{1/3} rho_0^{2/3} (M/M_{earth})^{2alpha/3}$$
$$ g = left(frac{4pi}{3}right)^{2/3} GM_{earth}^{1/3} rho_0^{2/3} (M/M_{earth})^{(2alpha+1)/3}$$
So, bar a (highly possible) algebraic slip, if you plot $log g$ vs $log M$, the gradient is $(2alpha+1)/3$, which from your plot, gives $alpha simeq 0.92$ - i.e. the average planet density increases almost linearly with mass.
The intercept then is
$$ b = log left[ left(frac{4pi}{3}right)^{2/3} GM_{earth}^{1/3} rho_0^{2/3}right],$$
which yields $rho_0 simeq 3.5$ kg/m$^3$ (NB: I subtracted 2 from your $b$ to make it SI; giving a density of around 814 kg/m$^3$ at a Jupiter mass).
The fact that density is almost proportional to mass can be found from the same dataset. e.g. see below. Below 0.1 Jupiter masses, the relationship appears to break down, though in actual fact very few of the densities for such planets are very accurately measured (since it requires a radius from a transit), but it works well enough in the range you have plotted. The physics here is that gas giants are governed by a partially (electron) degenerate equation of state that results in them all having a similar radius from about a tenth of a Jupiter mass to about 50 Jupiter masses (albeit with considerable and largely unexplained scatter). Thus the density is proportional to mass. This relationship does not work for small rocky planets, where the radius decreases for smaller masses. Thus your nice line in log space does not continue to lower masses (see plot at bottom - NB: some of the low-mass points have huge uncertainties).
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