Does the $\bf{1+3}$ representation of $SU(2)$ also represent $SU(2)\times SU(2)$?

Your group theory text probably betrayed you if it did not spend much time contrasting the two cases. A possibly related question is 254461. People use loose language and symbols that aggravate the confusion. Talking abstractly without explicit hands-on formulas clinches it (the confusion)!

Let me stick to your 4-dimensional matrices and vectors, all tensor products of 2x2 matrices and 2-vectors in both cases.

• A tensor-product (Cartesian product) group such as $SU(2)\times SU(2)$ has group elements of the type $\exp (i\theta^a \sigma_a) \times \exp (i\phi^b \tau_b)$, where I am using σ s and τ s for Pauli matrices acting on the left and right spaces, respectively---they are apples and oranges, think of them as spin rotations and isospin rotations. They are side by side. Importantly, their rotation angles are different and hopelessly unrelated, θ for space/spin rotations and φ for isospin rotations. You might combine them into a 4 dim space acted upon by 4×4 matrices, without absolutely any meaning. Crucially, you chose one of many possibilities for your group. The isospin group might have been supplanted by a different flavor group, say SU(3), so, then $SU(2)\times SU(3)$ and you'd have Gell-Mann λ s instead of τ s and your φ s would now be 8.

• This is very different from a Kronecker product of two representations of SU(2), so the same group, here chosen to be of two doublet representations: adding two spin 1/2 s. The group element may be acting on two different spaces, left and right, but with the same angles, like synchronized swimming, $~~~\exp (i\theta^a \sigma_a) \otimes \exp (i\theta^a \tau_a)$. You could have chosen different sized spaces for left and right, but the generators in the exponent should always be representation matrices of the same SU(2) in the representation of your choice, with the same angle. To see this, consider the coproduct in mathematese, namely $$ \Delta(J^a) = \sigma^a \otimes 1\!\!1 +1\!\!1\otimes \tau^a .$$ This coproduct is easy to see obeys the same Lia algebra as either representation, left or right, here chosen to be both Pauli matrices, "accidentally", by you. So then $ \Delta(J^a) $ is a fine representation of SU(2). Saturate it with three angles and exponentiate it to get the group element, $$\exp (i \theta^a \Delta(J^a) )= \exp (i \theta^a( \sigma^a \otimes 1\!\!1 +1\!\!1\otimes \tau^a))= \exp (i \theta^a \sigma^a)\otimes \exp (i \theta^a \tau^a), $$ where the left and right terms in the exponent commute among themselves, and so the exponential factors to two exponentials acting on the left and right spaces, respectively, in tandem. Now it turns out the 4-dim rep of the $\Delta(J^a)$s is reducible. That is, if we wrote it down as a 4×4 matrix, it would be reducible: that is, a suitable similarity transformation (Clebsch-Gordan) would transform it into a block matrix; here, kind of silly, since it would have a 3×3 block and a 1×1 block with zero entries, as the spin matrices for the singlet rep are 0s. The 3×3 block would be just the spin-one spin matrices, $j^a$. The group action on this rearranged 3+1 space would be $~~\exp(i\theta^a j^a)\oplus 1$, since exp(i θ 0)=1, quite exceptionally. Such would happen, of course, for all Kronecker products of any reps, not just your chosen example Pauli matrices. So here it does make sense to rearrange the tensor product space, as it facilitates the reduction of the reps.

• Takeaway: look at the angles---the transformation parameters: when you are combining groups, each group will have different ones, even if the two groups coincide. Now you are ready to face the Lorentz-group reps: different groups, many angles! If, instead, you are combining reps, the angles are the same, as many as the dimensionality of the group/Lie-algebra, but never more.

I guess what you are missing is the following:

given a representation $\rho(g)$ of $g\in$SU(2) acting on some vector space $V$. We define the representation $\rho_\otimes$ of SU(2) (not of SU(2)$\times$SU(2)) on $V\otimes V$ as $$\rho_\otimes(g) (v_1 \otimes v_2) = \rho(g) v_1 \otimes \rho(g) v_2.$$ So in fact we are defining the tensor product of two representations as a representation of SU(2).