OK, here's what I've been able to piece together with a little research (not really my field, so I may get a few things wrong).
The short answer is a combination of: A) It's pretty hard to actually disrupt a massive star via a neighboring supernova, and the two stars aren't intially that close to each other; and B) Lots of complicated effects due to the binary interaction.
I'll focus on what seems to be the best current model for forming a compact double neutron-star (DNS) system -- one with a small enough separation in the resulting neutron stars so that gravitational-wave emission can cause them to merge within a few billion years -- which is discussed in this 2017 article by Tauris et al.. It looks a bit like this:
Massive binary star system forms, with moderate separation (not as close as the final DNS binary will be).
Primary star (initially more massive) evolves faster, to the point where it expands and fill its Roche lobe, leading to mass transfer to the secondary.
This means the primary loses mass while the secondary gains mass. So when the primary reaches the supernova stage, it has less mass in its envelope to eject -- and the companion star may now be the more massive of the two.
Primary's core collapses, producing first SN. This probably won't destroy the companion; it might disrupt the system -- not so much because the companion is kicked a lot but because the NS remnant is lower mass than the pre-SN primary star, and it probably gets a kick from explosion asymmetries. But recall that the companion star may now be more massive, so the gravitational disruption is less.
Companion star evolves, expanding to overfill its Roche lobe, leading to a "common envelope" phase where the NS is actually inside the companion star's outer envelope.
Common envelope phase leads to
A. Significant shrinking of the orbit as the NS experiences dynamical friction;
B. Ejection of much of the companion star's outer envelope.
System is now much more compact, with a NS and a stripped helium-star companion.
Possible further mass transfer from companion onto neutron star, leading to even less of an envelope around the companion and an even lower mass.
Companion's core collapses, causing second SN. This has little effect on the first NS (i.e., the former primary star), since the NS is a very small target with extreme density and surface gravity: very hard to disrupt. In addition, the companion has very little envelope left to eject at this point.
Some key points:
The binary starts off with a significantly wider separation than the final DNS will have, so the effects of the first SN on the companion will be less extreme than you might imagine;
Mass transfer (and ejection) during the Roche overflow and common envelope stages removes mass from the stars, which means there's less envelope to eject when a SN kicks off (so it does less damage to the other star/NS) and also that the SN ejection of a star's envelope doesn't have a strong effect on the gravitational binding energy, since there's less of an envelope to eject.
This paper by Liu et al. (2015) presents some modern simulations of the effect of a SN blast wave on a nearby companion star. It turns out to be pretty hard to disrupt the companion: even for relatively small separations, the companion only loses about 10% of its mass, and this goes down to a few percent or less as the separation increases. (Remember that the companion is itself a massive star most likely still on the main sequence, and so relatively compact, with high surface gravity.)
Can the explosion push the companion star away? Probably not very much: again, the companion star is itself quite massive -- maybe more massive than the original primary because of the mass transfer in stage 2 (above) -- so it's hard to push it around.
