Is it possible for two Hopf algebras $H$ and $H'$ such that $H$ has skew-primitives(i.e. elements $x$ such that $\Delta(x)=g\otimes x+x\otimes h$ with $g,h$ grouplike elements) but $H'$ does not have skew-primitives, that $$\text{mod}(H)\cong \text{mod}(H')$$ as monoidal categories?
I asked this question a couple of days ago. The current question is an extreme case of that question.