Why a US dollar and a Euro are roughly the same value.

A US dollar and a Euro are roughly the same value: 1 USD is about 1 EUR, very roughly (rather than, say, 1 USD = 1000 ITL = 10^25 ZWD = 60 RUR).

There is history behind that which I've not seen presented in one place. Here is my vague understanding of it.

For a few decades before the early 1970s, the Bretton Woods system fixed USD and gold (at $35 = 1oz) and many other currencies were pegged to gold/USD.

This ended in 1971, when the USD would no longer freely converted into gold by the US government, and major world currencies became free floating.

Around that time, although at little bit before, the IMF Special Drawing Right came into existence. This was defined, at the start, to equal 1 USD, which it did until the end of Bretton Woods, but it wa. The only time I've really encountered this in real life was in small print on the back of aeroplane tickets, where compensation amounts were denominated in SDRs.

The SDR was composed of a basket of different world currencies (at time of writing, USD, EUR, CNY, JPY, GBP in defined ratios) so as the USD-value of those component currencies change, so does the USD-value of 1 SDR, rather than being fixed to the initial 1 SDR = 1 USD ratio. Over time, though, the exchange rate has stayed very roughly 1:1

In Europe, they invented the European Unit of Account which was a basket of specifically European currencies. This was scaled so that at the start, 1 unit of account was equal to 1 SDR (and so inherited the property of being roughly 1 USD). Being composed of a different based from the SDR, it varied in value with respect to both the SDR and the USD.

Next came the ECU (which was invented just before I was born), another basket of European currencies which replaced the European unit of account at 1:1

Finally, along came the Euro, an actual currency with paper notes and metal coins. This replaced the ECU again at 1:1, but again was a different basket: While the ECU basket had included GBP (British pounds), DKK (Danish crowns) and GRD (Greek Drachmas), the Euro did not include those.

So the Euro has the US dollar as its great-great-grandfather through a series of basket currencies, each branched from the previous at a 1:1 ratio.

Of course, all that 1:1 substition could have gone very differently: the Zimbabwe dollar was equivalent to a Rhodesian Dollar which was equivalent to half a Rhodesian Pound (as happened with most pound decimalisations) which replaced the Rhodesia and Nyasaland pound which replaced the Southern Rhodesian Pound, equivalent to a pound sterling - but the GBP:ZWD exchange rate at the very end was something like 1:10^25.