You seem to be conflating two thing:
1) What does correlation mean?
2) Can data mining and other issues mess this up?
Correlation between two variables means that the two variables are correlated: One tends to be higher when the other is higher and lower when the other is lower. Correlation may be due to some third variable, or it may not. It may be due to an outlier, or it may not, etc.
Correlation of time series (like your two stocks) is often due to a 3rd variable: Time. Stock prices tend to move in sync with each other.
And, if you "extensively data mine" then even random noise will produce some very strong correlations. If you look at, say, the correlations of 1000 stocks with each other, then you have 1000*999/500 correlations. You can see how many would be (say) above .9, even if all the prices were utterly from knowledge of the correlation coefficient's properties (standard error) or from simulation.
But if you look at those 500,000 correlations, you will see that they don't behave exactly like the random ones: they tend to be positive.
In case of non-linear correlation Spearman's Rank-correlation is one method
and one more method is called Kendall's Tau
R code for Spearman's rank correlation:
cor(X, Y ,method= "spearman")
R code for Kendall's rank correlation:
cor(X, Y ,method= "kendall")
Best Answer
The concordance correlation can be thought of as a measure of agreement. The question is: do two variables $x$ and $y$ (say) have identical values? If so, the concordance correlation will be 1. The question makes no sense unless the variables have the same units of measurement or more generally are recorded in the same way.
You can calculate a concordance correlation for any variables you like, but the answer will be of no use unless your question is about agreement. You could have a deterministic relation $y = \sin x$, but concordance between $y$ and $x$ will be a meaningless number, if only because concordance correlation does not adjust for different units.
For an informal introduction to this area, see
Cox, N.J. 2006. Assessing agreement of measurements and predictions in geomorphology. Geomorphology 76: 332-346. http://www.sciencedirect.com/science/article/pii/S0169555X05003740
Here "in geomorphology" indicates the field of the examples, not a restriction of statistical scope.