$(\frac{1}{2},\frac{1}{2})$ representation of $SU(2)\otimes SU(2)$

I) Firstly, we are talking about the direct or Cartesian product $SU(2)\times SU(2)$ of groups, not the tensor product$^1$ $SU(2)\otimes SU(2)$ of groups.

II) Secondly, $SU(2)\times SU(2)$ is not isomorphic to the Lorentz group $SO(3,1)$ but rather to a compact cousin

$$[SU(2)\times SU(2)]/\mathbb{Z}_2~\cong~ SO(4).$$

In particular, a $(\frac{1}{2},\frac{1}{2})$ irrep under $su(2)\oplus su(2)$ corresponds to a 4-dimensional fundamental vector representation under $o(4)$.

III) Thirdly, OP might be thinking of the complexified Lorentz group $SO(3,1;\mathbb{C})$, which has double cover $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$,

$$[SL(2,\mathbb{C})\times SL(2,\mathbb{C})]/\mathbb{Z}_2~\cong~ SO(3,1;\mathbb{C}).$$

cf. this Phys.SE post. In particular, a $(\frac{1}{2},\frac{1}{2})$ irrep under $sl(2,\mathbb{C})\oplus sl(2,\mathbb{C})$ corresponds to a 4-dimensional fundamental vector representation under $o(3,1;\mathbb{C})$.

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$^1$ Note that there exist various Abelian and non-Abelian tensor product constructions for groups. E.g. for the Abelian group $(\mathbb{R}^n,+)$, the tensor product is $\mathbb{R}^n\otimes\mathbb{R}^m\cong \mathbb{R}^{nm}$, while the Cartesian product is $\mathbb{R}^n\times\mathbb{R}^m\cong \mathbb{R}^{n+m}$.

The problem here is with the identification of the $(A,B)$ values of a representation with spin. $A$ and $B$ do not correspond to spin (they are not even Hermitian!), they just happen to obey $SU(2)$ Lie algebras, and as such they add up in the same way that spins do. When we say that $A_\mu,J_\mu,p_\mu,...$ are all in the $(\frac{1}{2},\frac{1}{2}) $ representation of the Lorentz group we mean that they transform as a four-vector, that's all. People may get lazy and say they are spin 1 objects, but what they really mean is $(A,B)$ spin 1 objects.