Does it really matter if a correlation is spurious? Let’s say you are trying to find if there is a correlation between two stock prices, where both are likely non stationary series. You have no concern as it relates to a potentially causal relationship...
You run a simple correlation analysis against all the rules. Both our series are autocorrelated and non stationary. You find there is a 98% correlation so you conclude they depend on each other.
This is the conversation I just had with a colleague... but I think they are 100% wrong and I’d like some confirmation.
If you find two autocorrelated and non stationary series to be 98% correlated, then the correlation is likely spurious. What this means to me is that the correlation we observe is likely due to complete chance (and their correlation is likely a result of their mutual dependence on something else outside of the two series themselves). So if our goal is to identify the extent to which these two series “depend” on each other, finding a valid correlation coefficient is necessary. Is this correct?

The whole notion of "spurious" correlation is easy to misinterpret.  Correlation is correlation --- if it is estimated well (i.e., via a good estimator and with a reasonable amount of data) then we can confidently say that the correlation is such-and-such.  Correlation is a statistical measure with an extremely weak interpretation --- it just measures the tendency for things to vary together (usually measured linearly), irrespective of the cause of this tendency.  The only thing that can be spurious is if we go further than this and interpret the correlation in a way that is not justified.  This can occur if a person uses correlation to infer a causal relationship between variables, or it can occur if a person uses marginal correlation to infer conditional correlation.  In either case the larger inference can be "spurious" insofar as it does not follow from the correlation.  As I've noted in another answer, I've always hated the term "spurious correlation" because it it is not the correlation that is spurious, but the inference to some stronger result.  If it were up to me we would never use this term, and would instead just state what we actually mean --- e.g., "spurious inference of cause", "spurious inference to conditional correlation", etc.
Now, with that little rant out of the way, let me address your specific concern.  Since you are only interested in describing the past statistical relationships between the stock prices (as you say in your comments), you can report the correlation, but it should come with a number of important caveats on interpretation.  Firstly, you should note that strong correlation between time-series can occur even for purely deterministic series with no statistical variation, so it often does not reflect any stochastic dependency between the series.  This is something that has been recognised in the statistical community for over a century (see e.g., Yule 1926 and see this related answer).  Secondly, even if changes in the stock prices are correlated, the ability to predict one stock from the other will depend on the cross-correlation in changes in the stock prices at sufficient lag values to allow use of one series to predict changes in the other.   In large part, analysis of stock prices is best done by looking at lagged cross-correlation of changes in price, rather than correlation of the price series themselves.
