So the Eidolon can take an evolution called No Breath. What if the summoner already has a similar ability, where they don't breathe and can't be affected by anything that needs to be inhaled? When fused, could they then be poisoned? Would they need to breathe when fused even though their real body doesn't need to? Could the fused summoner/eidolon suffocate or drown when the summoner can't?
A synthesist has no separate eidolon to harm
If the summoner possessing the archetype synthesist is immune to an effect, he's also immune to that effect while his eidolon is present. When synthesist and summoner are fused, there's no distinction between summoner and eidolon. The two are one creature. Among other effects, this means that
…[t]he synthesist gains the eidolon’s hit points as temporary hit points. When these hit points reach 0, the eidolon is killed and sent back to its home plane.… The synthesist also gains access to the eidolon’s special abilities and the eidolon’s evolutions.… Neither the synthesist nor his eidolon can be targeted separately, as they are fused into one creature. The synthesist and eidolon cannot take separate actions. While fused with his eidolon, the synthesist can use all of his own abilities and gear, except for his armor.…
Typically, the only things that affect the fused eidolon exclusively instead of the synthesist–eidolon fusion creature are effects that are like the spells banishment and dismissal. The rest of the time it's just synthesist, synthesist, synthesist, not, as is normally the case, summoner and eidolon as separate beings.
A synthesist isn't the invincible Iron Man nor the incredible Hulk, but a uniquely weird fantasy creation. A sythesist activates his eidolon, changes his appearance, and gains a suite of special abilities. When the temporary hp are depleted, the creature's appearance returns to normal, and it loses that suite of special abilities.
While fused with his eidolon, an assassin could no more poison a synthesist's eidolon separately than the assassin could opt to poison just a victim's skin or liver. Likewise, While fused with his eidolon, a synthesist can drown, but the fused eidolon can't also or separately drown—they aren't a they but one being.