It all seems unnecessary either way as you don't leave the airport and then you go through passport control when you arrive in the next country again anyway!
OP will not go through passport control. So there ought to be no confusion.
You always go through passport control when you enter a country (or, in the case of Schengen, one of its member countries). OP is leaving the UK and entering Peru. She will remain airside at Amsterdam and not have to show her passport.
Also not helpful having a UK passport at all - I am mid-way applying for my Irish one to help speed travel up!
Having an Irish passport would not change anything in the OP's situation. If OP had an EU passport and was to go landside (through immigration) in Amsterdam - which, again, she is not doing - she would still have to go to through passport control. The only difference would be that she could use the EU lane (and, from late 2026, she would not need an ETIAS). EU citizens always have to show a passport (or an ID card) when they cross the Schengen external border, because they have to prove that they are indeed EU citizens.
I absolutely did have to in November. Travelled from Birmingham UK to Schiphol, onwards to Austria and I was surprised there was passport control again
Some Schengen countries are currently making a big show of checking passports even for internal Schengen flights, to show their voters how big and manly and tough on brown people they are. France did this for over three years after the Bataclan attacks of November 2015, even though the terrorists drove across the open border with Belgium where there are no checks (a lot of the crossings are country lanes). So what probably happened is that after you entered Schengen at Amsterdam and got on an internal Schengen flight, which should not have led to passport control in Austria, you encountered this extra level of check because someone in the Austrian government was scared of the far-right (or perhaps was an actual member of the far-right).