Suppose $H$ and $H'$ are two (possibly infinity dimensional) Hopf algebras which are not isomorphic as Hopf algebras, but are isomorphic as algebras. More specifically they are not isomorphic as Hopf algebras because the (skew)-primitives behave differently in both Hopf algebras. It is well-known that one can detect skew-primitives in the categories of comodules, indeed, skew-primitives are determined by extensions of one dimensional comodules and vice-versa.
However, can one detect skew-primitives in the representation category of $H$? Note that Tannaka-Krein reconstruction might not be a good approach as the Hopf algebras are infinite dimensional.
Edit: Let $H$ be a Hopf algebra and $x\in H$ a skew-primitive element. Let $K$ be the Hopf-subalgebra generated by $x$. The embedding $K\hookrightarrow H$ induces a monoidal (restriction) functor $$\text{mod}(H)\rightarrow \text{mod}(K).$$ Is there any hope of classifying such functors given that you know the structure of $H$ and $K$? Perhaps, skew-primitives can be translated to such easy restriction functors?
Suppose $F:\text{mod}(H')\xrightarrow{\sim} \text{mod}(H)$ as monoidal categories, then by Eilenberg-Watts' theorem $F\cong -\otimes_{H'}N $ where $N$ is a $H'$-$H-$bimodule. (We didn't use that $F$ is monoidal to invoke this theorem, the fact that $F$ is monoidal yields extra structure on $N$, but I'm not sure how much structure). Thus composing $F$ with the embedding $\text{mod}(H)\hookrightarrow \text{mod}(K)$ yields a monoidal functor $\text{mod}(H')\rightarrow \text{mod}(K)$. Is this functor still a 'restriction functor'? If so, that would be interesting since $H'$ might not have skew-primitive elements and such functors might not exist.