Why is p-value termed as P(Data | Hypothesis/Model)? As the title suggests, Why is p-value termed as P(Data | Hypothesis/Model) and not P(Hypothesis | Data)? Shouldn't both be the same? Why is P(Data | Hypothesis) != P(Hypothesis | Data)? Is there any logical reasoning here that I am missing?
 A: The equivalence that you propose represents a fundamental (and often made) error, which the American Statistical Association has been trying to stamp out for some time now. See the statement by Wasserstein, Schirm & Lazar (2019).
You have one data set, but there are multiple competing hypotheses. The probability one might assign to one given hypothesis ought to reflect the relative strength of evidence for each hypothesis relative to the competitors. A p-value is nothing like that, in itself.
When you look at a hypothesis test generating a low p-value, it can seem plausible that the p-value represents the probability of the hypothesis. But think more generally. Imagine a hypothesis test involving a simple hypothesis with one parameter. That parameter is continuous in value, so the value lies on a continuum. There is a p-value that goes with every single one of the infinity of different values on that continuum, and some of those p-values would be quite high, very close to 1.0. The continuum collectively represents all possibilities for the value of the parameter. When we sum across all probabilities for an event, the sum should be 1. But summing all of these p-values will produce a number much, much larger than 1. Therefore, the p-values are not the probabilities that each different value of the parameter is correct or true.
A: The literal definition of a p-value is the probability of finding a test statistic at least as extreme as the observed test statistic, given that the null is true: something like (but not quite the same as) $P(\text{Data}\vert\text{Null is true})$. If such a result is unlikely (low p-value), we see that as evidence against the null hypothesis. Sufficiently strong evidence against the null hypothesis make us say, “No, we’re unlikely to get this result if the null is true, but we did get this result, so the null sure seems false,” kind of a proof by contradiction.
You claim that a p-value measures the probability of your null hypothesis given the data you’ve collected. This is a common mistake to make, but it is indeed a mistake. 
