Correlation: A Magnitude or Power quantity?
There are different ways to look at this.
In general, it's useful to think of the correlation as a power quantity.

If you slog through the units, the correlation will have power-like units: $V^2$ for example
The Fourier Transform of the autocorrelation is the Power Spectral Density, which is clearly a power-like quantity

So what's happening in your example? Let's assume that we have $y = x + n$ where $x$ and $n$ are mean-free, uncorrelated and have the variances $P_x = \sigma_x^2 = &lt;x^2&gt;$ and $P_n = \sigma_n^2 = &lt;n^2&gt;$. Since they are uncorrelated we have $P_y = P_x+P_n $
The SNR would be $$SNR = 10log_{10}\frac{P_x }{P_n}$$
The argument to the log is a ratio of powers.
The covariance is
$cov(x,y) = &lt;x\cdot (x+n)&gt; = P_x$ which has power-like units. The normalized correlation coefficient comes to be.
$$\rho = \frac{cov(x,y)}{\sqrt{{P_x \cdot P_y }} = \frac{P_x}{\sqrt{P_x \cdot (P_x + P_n) }} $$
$$\rho = \frac{cov(x,y)}{\sqrt{P_x \cdot P_y }} = \frac{P_x}{\sqrt{P_x \cdot (P_x + P_n) }} $$
So that's also a ratio of powers: cross-power to geometric mean of the signal powers. Since it's a ratio it's unitless. Let's square it:
$$\rho^2 =\frac{P_x^2}{P_x \cdot (P_x + P_n) } = \frac{P_x}{P_x + P_n } $$
The most reasonable physical interpretation of this is STILL a ratio powers: signal power to signal power plus noise power. Since the ratio is unitless, the square of ratio is still unitless, so there is no direct conflict here.