In my opinion, the source that you link to is wrong in that it is confusing conditioning with assumptions.
Fisher's exact test conditions on the margin totals, meaning that it does not use any information about independence that might be inferable from the margin totals. In probability theory, once you condition on a random variable, the random variable is then treated as fixed in the downstream computations. This however is purely a mathematical device. It is not at all the same as assuming that the margin totals were fixed in advance as part of the experimental design. It is simply a matter of choosing what information to extract from the data.
Conditioning actually reduces the assumptions made by the test. Since Fisher's test only uses the distribution of the cell counts given the margin counts, it therefore makes no assumptions about how the margin counts were generated.
The source you link to states that Fisher's exact test makes more assumptions than Pearson's chisquare test or the G-test. In my opinion, this statement is wrong. All three tests condition on the margin totals and all three make the same distributional assumptions. Fisher's exact test gives correct control of the type I error rate regardless of whether the marginal totals are fixed or not and the p-values are exact in the sense that they are derived from exact computations using the hypergeometric distribution.
IMO there is a lot of misunderstanding about this issue on the internet and it possibly derives from the question of optimality. If the margin totals are not fixed by the experimental design, then potentially there is information about independence contained in the row and column totals, and hence potentially one might be able to create a statistical test that is more powerful than Fisher's exact test. There is some interesting literature about this but in practice (i) there's not much information to retrieve if the counts are small and (ii) Fisher's exact test is already powerful enough when the counts are large.
Another misunderstanding is that people interpret "exact" to mean that the type I error rate is controlled exactly at a specified rate. Like all tests that return exact p-values for discrete counts, Fisher's test gives somewhat conservative control of any pre-specified type I error rate, simply because the p-values are granular (and this has nothing to do with whether the margins are fixed or not).