The Bitcoin network doesn't authenticate users. The Bitcoin network doesn't know or need to keep track of people's identities.
Just as a recipient of a $10 bill or €50 banknote don't need to know the identity of the other person, they just need to examine the paper to see if it looks genuine.
Bitcoin is cash, not a bank account.
Bitcoin transactions contain a cryptographic proof that the spender is entitled to spend an amount of Bitcoin that is the output of some prior transaction. This proof is based on private-key cryptography where a private key can be used to produce a cryptograhic signature of data that proves the author knows a private key corresponding to a public key associated with the amount. It is a secret number which you can mathematically prove you know without revealing the number.
The private key is chosen essentially at random. There are so many possible private keys that there is no realistic chance of two people randomly choosing the same key anytime in the expected lifetime of the Universe.
How to use cash?
Lets say you are on vacation in France, you see a cake you'd like to buy in a Boulangerie. The vendor looks at your $10 bill and says "I only take Euros". What do you do? You walk to a Bureau de Change show them your $10 bill and ask "How many Euros will you give me for this?". If you like their answer you get Euros you can use to buy your cake. If you don't like their answer, maybe you ask a passerby who looks friendly, maybe you ask in your hotel, maybe you can think of a thousand other ways to convert cash in one currency to cash in a different currency without the direct personal involvement of governments, mints or central banks?
Remember that people traded goods between countries long before computers, long before electricity. Swapping cash in Dollars for Sterling yesterday is fundamentally no more magical than swapping crowns for doubloons centuries ago or swapping Yen for Bitcoin today.
Much the same applies to any currencies. Including Bitcoin, cowrie shells or chunks of gold amulets. The business of money-changer is an ancient one but even they do not have a monopoly and do not prevent two people directly swapping things of equivalent value.