I wonder if it is a case of saying 'covariance' in a different language.

One could consider two random variables $X$ and $Y$, two vectors are then the realizations of these variables, which can be written as sets of measurements $\{X_k\}_{k=1\dots N}$, $\{Y_k\}_{k=1\dots N}$, or as vectors $\mathbf{X}=\left(X_1,\,\dots X_n\right)^T$, $\mathbf{Y}=\left(Y_1,\,\dots Y_n\right)^T$.

The dot product between the two vectors is then related to sample covariance $S^2_{XY}$:

$$ \begin{align} \mathbf{X}^T.\mathbf{Y}&=\sum_{k=1}^N X_k Y_k=\left(N-1\right)\cdot\frac{1}{N-1}\sum_{k=1}^N \left(X_k-\bar{X}\right) \left(Y_k-\bar{Y}\right)+N\cdot\bar{X}\cdot\bar{Y} \\ &=\left(N-1\right)\cdot S^2_{XY}+N\cdot\bar{X}\cdot\bar{Y} \end{align} $$

Where $\bar{X}$ and $\bar{Y}$ are sample means ($\bar{X}=\sum_k^N X_k/N$ and same for $Y$).

If the samples are conditioned so that sample means are zero (e.g. z-scaling), then dot product is proportional to the sample variance, which is an estimator of the variance between the two variables.

If $X$ and $Y$ are statistically independent then their covariance will be zero. Furthermore, if the two variables come from suitable distributions, e.g. Normal distribution, then zero variance actually implies statistical independence