This is an interesting flight. Unlike SQ52/SQ26, MXP-BCN is a flight within the EU and an intra-Schengen flight. With SQ25/26 or SQ51/SQ52, all passengers disembark at FRA/MAN. Those connecting onto JFK and IAH enter the airside terminal whereas those with the final destination FRA/MAN directly proceed to passport control. As a result, airport staff check boarding passes of all passengers to ensure no security and immigration concerns.

With SQ378, I would believe those continuing onto BCN wouldn't be allowed (and aren't allowed) to disembark at MXP because the aircraft gate would be airside in the Schengen terminal at MXP. This flight does have fifth freedom rights so there are passengers flying only from MXP to BCN. Passengers flying all the way to BCN wouldn't be allowed to disembark at MXP because they wouldn't have cleared Schengen immigration yet. As a result, only those with final destination MXP disembark and proceed to immigration. Those traveling only from MXP to BCN use the Schengen terminal/gate to board SQ378 to BCN without having to clear immigration.

This works somewhat similarly to SQ478 (SIN-JNB-CPT) apart from the fact that the flight doesn't have eighth freedom rights. Therefore, all passengers board from SIN and no one joins in from JNB. Passengers either disembark at JNB or CPT depending on their final destination. Those continuing onto CPT stay onboard at JNB similar to those continuing to BCN at MXP.

What is interesting is how it works on SQ377. I saw a Youtube video of someone who got caught in a mess with EU immigration after passport control at BCN stamped him out of the Schengen region even though he was only flying BCN-MXP. Would be interesting to hear how it works on that leg. How does passport control at BCN work?