If we connect $S^{1}_a\times S^1_b$ to $S^1_c\times S^1_d$ by identifying $S^1_a\times x_0$ with $S^{1}_c\times x_1$ to form $T'$ is $\pi_1(T')=\left<a,b,c|aba^{-1}b^{-1},aca^{-1}c^{-1}\right>$. This seems correct as $$\pi_1(S^{1}_a\times S^1_b)=\left<a,b|aba^{-1}b^{-1}\right>,\pi_1(S^1_c\times S^1_d)=\left<c,d|cdc^{-1}d^{-1}\right>,$$ so natural by Siefert van Kampen we have $$\pi_1(T')=\left<a,b,c,d|aba^{-1}b^{-1},cdc^{-1}d^{-1},\text{ amalgamation relations}\right>,$$ hence we just have to find the amagamation relations. We know $\pi_1(S^1_a\times S^1_b\cap S^1_c\times S^1_d)=\left<e|\text{no relations}\right>,$ so we just have to find the image of $e$ in the respective copy of our Tori and identify them. Taking $\pi_1(S^1_a\times S^1_b)$ to have basepoint $y_0\times x_0$ and $\pi_1(S^1_c\times S^1_d)$ to have basepoint $y_1\times x_1$ it seems naturally that upto homotopy $e$ is identified with a generator of each fundamental group. That is $i_1(e)\in\{a,b\}$ and $i_2(e)\in\{c,d\}$ if $i_1,i_2$ are the homomorphisms induced by inclusion. Since $\{a,b\}$ are symmetry why not take $i_1(e)=a$ and similarly $i_2(e)=d,$ then $$\pi_1(T')=\left<a,b,c,d|aba^{-1}b^{-1},cdc^{-1}d^{-1},da^{-1}\right>.$$ The relation $da^{-1}$ tells us $da^{-1}=1,$ hence $d=a$, so can't we reduce $\pi_1(T')$ to $\left<a,b,c|aba^{-1}b^{-1},aca^{-1}c^{-1}\right>?$
Now I do realize that Siefert van Kampen applies to open sets, and not all the sets I listed were open in $T'$. However, we can just enlarge them by including small open collar like neighborhoods. In which case our enlarged open sets deformation retract to the ones I was working with.