assumptionscorrelationpearson-r
What are the assumptions for the proper use and interpretation of the Pearson's correlation coefficient?
How is $\DeclareMathOperator{\cov}{cov}\cov(X,Y)$ affected by (mild) deviations of linearity?
How is $\cov(X,Y)$ influenced by the presence of heteroscedasticity?
What matter is $cov(X,Y)$. Denominator $\sqrt{var(X)var(Y)}$ is for getting rid of units of measure (if say $X$ is measured in meters and $Y$ in kilograms then $cov(X,Y)$ is measured in meter-kilograms which is hard to comprehend) and for standardization ($cor(X,Y)$ lies between -1 and 1 whatever variable values you have).
Now back to $cov(X,Y)$. This shows how variables vary together about their means, hence co-variance. Let us take an example.
Lines are drawn at sample means $\bar X$ and $\bar Y$. The points in the upper right corner are where both $X_i$ and $Y_i$ are above their means and so both $(X_i-\bar X)$ and $(Y_i-\bar Y)$ are positive. The points in the lower left corner are below their means. In both cases product $(X_i-\bar X)(Y_i-\bar Y)$ is positive. On the contrary upper left and lower right are areas where this product is negative.
Now when computing covariance $cov(X,Y)=\frac1{n-1}\sum_{i=1}^n(X_i-\bar X)(Y_i-\bar Y)$ in this example points that give positive products $(X_i-\bar X)(Y_i-\bar Y)$ dominate, resulting positive covariance. This covariance is bigger when points are aligned closer to an imaginable line crossing the point $(\bar X,\bar Y)$.
As a last note, covariance shows only the strength of a linear relationship. If relationship is non linear, covariance is not able to detect it.
If you have $n$ objects you will have $\frac{n(n-1)}2$ distances. So, unless $n<4$, you always have more distances than objects. This artificially inflates the number of independent samples when using Pearson's correlation.
Best Answer
The only real assumption of Pearson's correlation is that the variables are interval level. There are additional assumptions for tests of whether the correlation is 0, but the correlation is the correlation.
However, the correlation only examines the linear relationship between X and Y. So, while the correlation doesn't assume anything about the variables, it can be misleading in some cases and for some purposes. See the Anscombe Quartet for some extreme examples.
It is similar to the case with the mean: The arithmetic mean doesn't assume anything about the variables except, again, that they are interval. But the mean can be a misleading choice of measure for central location in some cases.
However, it is not necessarily the case that the mean or correlation are poor choices even with oddly distributed data: It depends on what you are trying to measure.
The central lesson is that it is always good to graph your data first. As Yogi Berra said "You can see a lot by looking"