Hi Thorben.
It's confusing because (I think) there are two types of "multiple testing" involved if you are doing ANOVA for many gene in more than two groups.
The first type, is that you are not testing 1 gene but many genes, like 20000. So here you must use multiple test correction to control the FDR.
The second type of multiple testing is if you have more than two groups (for a single gene). Then (strictly speaking) you need to do another multiple testing correction because you need to do a post-hoc test to see which groups are significant of all the group combinations.
As said, we don't do testing of multiple groups (only two groups), but of course we have multiple genes. so we need to correct for the first type of multiplicity. This is independent of DeSeq2, edgeR or LIMMA.
Hope this helps,
Ivo