Is maximum matching problem equivalent to maximum independent set problem in its dual graph?
To start with possible NP-hardness (where for each problem, we want a matching/independent set of size at least $k$):


Independent set is NP-hard on "normal" graphs (and also on hypergraphs)
Maximum matching is polynomial-time solvable on "normal" graphs, see the wikipedia page on matching.
Maximum matching is NP-hard in hypergraphs (as shown in this wikipedia page, it is even hard for hypergraphs where each edge contains only 3 vertices).


I believe both problems are equivalent in the following sense:  set $S \subseteq E(H)$ is a matching in $H$, if and only if $S$ forms an independent set in $H^*$. 

(If you need further explanation or if this is not your definition of equivalance, please clarify)