To what extent are lepton and quark generations tied in the Standard Model?

Is there some explicit link with generation-specific interactions? Or is it just a coincidence that there are three rungs in both ladders with increasing mass on both?

In the SM, there are no generation-specific interactions, beyond color, which factors out. The generation tabulation system simply arrays states according to increasing mass, a still mysterious pattern; which is why, to date, the neutrino mass eigenstates $\nu_{1,2,3}$ have not been assigned to generations, yet, pending complete experimental confirmation of the normal hierarchy/ordering (the $\nu_{e,\mu,\tau}$ are not mass eigenstates, and were on that chart to simply confuse and abuse; mercifully, they are going away). The increasing mass pattern is not a coincidence: it is the construction principle.

The weak interactions, through weak mixing, hop across all generations, and neutrinos mix a lot, unlike quarks, whose mixing angles are small. So scrambling green columns will do nothing to physics, as long as you take care to rewrite your PMNs matrix to reflect the labelling change. Of course you do need the number of states you see on such tables to be matched, 3 rungs to 3 rungs, whatever their order, so as to prevent gauge anomalies, invalidating gauge invariance, but the assignment of states in generations, the order of the rungs, is basically arbitrary.

In speculative GUTs (like the Georgi-Glashow SU(5)) one tries to link fermion masses (cf (22), since leptons and quarks are put in common representations, 5 and 10; but, again, alternate inequivalent models wiggle the leptons around associating them to different quarks, to adapt to negative proton decay results. So, indeed, such alternative speculative models take advantage of the SM freedom to scramble green columns.