My understanding of ordinal data is that it has a scale to it. For example: Happiness on a scale of 1-10. Economic Stability on a scale of 1-10. This can later be ordered to perform statistical analysis such as Spearman's correlation coefficient
Similarly continuous data is something that can also be measured and sorted, such as temperature of the room, evaporation rate etc. Which is quite similar to ordinal data.
Why is it that Pearson's correlation coefficient can provide viable results for continuous data and not for ordinal data, which can be ranked quite similarly?
Sure it can be used on ordinal data. For instance, an area of research in economics is the relationship between income (continuous variable) and happiness or psychological well-being (an ordinal variable). The difference is that Pearson’s measures linear relationship while Spearman’s measures any monotonic relationship, as discussed here.