I am currently interested in the correlation v.s. causation discussion.
I know that correlation might be spurious (see that spurious correlation website).
However, as the causality is inferred from statistics, isn't there a chance that the causality might be spurious too? How should I generally interpret that? (is it just like any other tests? that is being confident in causation up to a certain level)
The short answer is yes. In fact, "spurious causation" is a much bigger and more widely studied problem than "spurious correlation." It's just not usually called that.
Spurious correlation is a very particular kind of statistical fallacy that comes from testing a ton of different variables until you happen to find two that are correlated just by chance. It is a big problem, and can produce bogus causal claims. But if you test the correlation between the same two variables in a different dataset, you won't find a correlation there. Hence the correlation is spurious. But there are lots of OTHER reasons why a causal claim might be "spurious."
In fact when people say things like "correlation is not causation" what they are usually talking about the fact that people often claim that A causes B just because A and B are correlated. This can be true even if the correlation between A and B is not "spurious." The entire field of "causal inference" exists to try and understand when, and under what circumstances we can infer causal relationships based on correlational data.
For example, ice cream consumption and shark attacks are highly correlated. This is NOT a spurious correlation: you can (probably) find a high correlation between these two variables in many different datasets. But this still doesn't mean that eating more ice cream causes shark attacks. Instead the (very real) correlation between these two variables comes from a third "confounder" variable (high temperatures) which causes BOTH ice cream consumption and going to the beach, which in turn causes shark attacks.
One of the things that causal inference as a field teaches us is that causation can't purely be "inferred" from statistics. There is no statistical or mathematical test that, on its own, can tell us for sure what the relationship between A and B "means" in causal terms. To figure that out we always need to bring some additional information to bear: Was A measured before or after B? Was A "randomly assigned?" Are there any other variables that are correlated with both A and B that we have not measured? Does it make theoretical "sense" for B to cause A? These are really hard questions to figure out (search for "selection bias," "history bias," "reciprocal causation," "confounding variables," for some examples of these issues), and if you get them wrong, you might get the wrong causal story, even if the correlation between A and B is not "spurious."
Statistical analyses can help us make better causal inferences from correlational data, but these analyses always involve bringing in non-statistical assumptions or knowledge (even the approach of "causal discovery," which seems at times like it is claiming to uncover causal relationships through purely correlational analyses, still depends on a lot of very strong theoretical assumptions about what causal pathways are possible and what ones are not).
