How is two way authentication so instantaneous?

Sorry, I realise this probably a complete noob question. But I've spent so much time trying to find an answer and at this point not even sure if my keywords are right. Please please guide :(

Here goes -

How is two-way authentication so instantaneous? We are having the age-old push vs fetch debate regarding downloading certain time-sensitive updates in our mobile app. 'Fetch is better' team claims that there can be potential delays with push notifications via FCM. Additionally, they also claim that our android app can essentially do via frequent fetch what the web job detecting changes for push notifications will do so why implement all that. Are we stuck in 2011? probably :(

All this made me curious about two-way authentication from Microsoft/ banks approving transactions. Are they implemented differently to REST API? How are they so instantaneous? When I search anything it just tells me how to set it up. Can you please advise how it is architected etc or keywords to discover ourselves? Thank you in advance.

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