I'm somewhat more positive than the other two answers, but it all depends on (a) how your data actually look like and (b) what you will then do with the correlations, and whether there are good alternatives for doing this.
In any case I think that the sample correlation coefficient can be used as a descriptive statistic formalising the tendency in the data to show to what extent large values in $x$ tend to go together with large values in $y$ without the ambition of estimating any assumed underlying truth at high accuracy. As such it doesn't look wrong to me to use it to "summarize what I see" as you put it. Of course if you summarise your data in this way, you focus on a certain aspect and lose other aspects of your data, and you should ask what you lose and to what extent this would be relevant to your aims. Also it'd need a justification to represent any of these datasets by a single statistic in a later analysis, but I can well imagine situations in which this makes sense. Also it depends to some extent on your data (outliers are bad for Pearson correlation, and if you have some clear nonlinear patterns, these may not be well reflected either).
The other answers are correct stating that sample correlation on 10 data points isn't very precise estimating a "true underlying" correlation value, but I don't necessarily think that this would be required for "summarising" the data with potential use of the summaries in another analysis at higher level.