What is the exact relation between $\mathrm{SU(3)}$ flavour symmetry and the Gell-Mann–Nishijima relation

The Gell-Mann–Nishijima relation arises from electroweak symmetry breaking. If we vev our Higgs SU(2) doublet, $$ \langle(\phi^+, \phi^0)\rangle = (0, v/\sqrt{2}), $$ we find that the theory remains invariant under a combination of the diagonal, Cartan SU(2) generator, the weak hypercharge, and the hypercharge, $Y$, because $$ e^{iQ} \langle(\phi^+, \phi^0)\rangle = \langle(\phi^+, \phi^0)\rangle\\ Q \langle(\phi^+, \phi^0)\rangle = (T_3 + Y/2) \langle(\phi^+, \phi^0)\rangle = 0 $$ because on our Higgs doublet, $$ T_3 + Y/2 = \left(\begin{array}{cc} 1/2 & 0\\ 0 & -1/2\end{array}\right) + \left(\begin{array}{cc} 1/2 & 0\\ 0 & 1/2\end{array}\right) = \left(\begin{array}{cc} 1 & 0\\ 0 & 0\end{array}\right) $$ One can always find a combination of the U(1) and Cartan generator of SU(2) which annihilates the vacuum. This form of the relation and these hypercharge assignments are a convention. The general form is $Q=T+aY$, with $a$ determined from the the hypercharge of the Higgs boson.

Indeed, the formula only appeared empirically in 1956, before the eightfold way, for hadrons, long before quarks; and was seen to be such a basic fact that it informed the way flavor SU(3) was put together; and was subsequently spatchcocked into the gauge sector of the EW theory a decade after that--hence the alarming asymmetry of the hyper charge values.

Its basic point is that isomultiplets entail laddering of charge, $I_3$ being traceless, but in the early days of flavor physics, with just the strange quark, an isosinglet required its charge to be read by something, and so was incorporated into the 3rd component of Gell-Mann's diagonal $\lambda_8$, providing the needed 2nd element of its Cartan sub algebra.

Note that, in left-right flavor physics, say after the introduction of the charmed quark, C came as an addition to the strangeness, additively in the hypercharge, so (S+C+B)/2, whereas in the left-handed sector of the EW theory charm and strangeness (and T and B-ness) are in weak isodoublets, having escaped the hypercharge pen!