Let $\mathfrak{g}$ and $\mathfrak{g}'$ be Lie algebras. It is known that if $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as associative algebras, then it is not necessarily true that $\mathfrak{g}\cong \mathfrak{g}'$ as Lie algebras.

I am looking for examples such that $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as algebras but $\mathfrak{g}\not\cong \mathfrak{g}'$ as Lie algebras (over an algebraically closed field). Moreover, are there examples such that the categories $U(\mathfrak{g})-\text{Mod}$ and $U(\mathfrak{g}')-\text{Mod}$ are not monoidally equivalent?

I'm not very familiar with the isomorphism problem for enveloping algebras, a quick google search only gave me counterexamples in positive characteristic. I'd be very happy with examples in characteristic zero (infinite dimensions are allowed). I'm more into the monoidal stuff and might figure out myself whether the representation categories are monoidally equivalent.

**Edit:** I'm asking this because I naturally encountered a quantized version of this problem. Obviously the categories $U(\mathfrak{g})-\text{Mod}$ and $U(\mathfrak{g}')-\text{Mod}$ are Morita equivalent but there is more information here. First of all $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as algebras which clearly is stronger but they are also enveloping algebras of Lie algebras, further restricting possibilities. In the quantized version I'm looking at, I suspect the representation rings of both categories to be the same making the difference in the monoidal structure very subtle. So I'm wondering whether anything on this subject is known in the non-quantized world.