Everything is a measure of value. Right now we've practiced to value bitcoin equivalent to fiat. Even a satoshi is valued in fiat. From the beginning if the price is measured of its own, now everyone could've followed the same. The reality, bitcoin couldn't have reached this popular. Valuation in terms of fiat made people understand the value of bitcoin in a better way and the same made people focus on it.

Another thing bitcoin is completely on technology parameters, whereas with fiat there'll be gold reserves that back the fiat system. In this way bitcoin doesn't have the real backing. Maybe in the long run if there is official acceptance all around the world bitcoin can be valued of its own.

Bitcoin is usually compared to USD, may be occasionally with other fiat currencies. Do you realise there are other currencies apart from US dollar but we usually measure all these other currencies relative to the US dollar, that doesn't make other currency less valuable and same applies to bitcoin. The reason USD serves as a unit of measurement is simple, most international transaction are anchored around the US dollar. In short, the chances of bitcoin to stand alone without being compared to currencies like USD looks unrealistic in my opinion, so yes, fiat like the USD will exist to explain.

It is also important to note that "a=b" is thesame as "b=a" so we can as well ask the question the other way round; will bitcoin always exist to explain? So long as bitcoin is still a valuable crypto, the answer is Yes. USD(fiats) and BTC are like two different planet and as a bitcoiner, if you are in the world of USD you will have to convert some of your BTC to USD to be able to coexist with the dollar driven world, their is no way around it because the government of the world do not prioritise decentralisation practices and to convert it you'll need to relate bitcoin to USD (e.g: 1 btc = $64k), so its just a natural way it must be.

If I wanted to use Bitcoin as payment for my business. I wouldn't be surprised if someone wanted to give me Satoshi in exchange for Bitcoin but rather if they gave me $50 in exchange for Bitcoin. However many may say that paper money or fiat currency has been devalued since the arrival of Bitcoin but in reality, only some limited aspects of all these currencies, such as inflation, have automatically devalued fiat currency since the arrival of Bitcoin. All currencies that have unlimited supply are sure to devalue in the future. However, in this case, the comparison of Bitcoin with fiat currency will remain for ages.

It's almost impossible for Bitcoin to become the standard unit of account (or its own unit of value, per your OP). The world is still stuck with Fiat. Why? Because banks and governments are relevant. The only way this would work is if they're out of the picture for good. Not going to happen now or in another century.

Besides, people want something that's easy for them to understand. It's much easier to handle Fiat, than Bitcoin. BTC is divisible to 8 units, while Fiat has only two decimal points. Most wages are paid in Fiat, too. Which is why, the world will continue to use Fiat as the standard unit of account. At least, BTC makes a great alternative. What more can we ask for?

As long as Bitcoin exists, then there will always be a fiat currency as a tool to know or measure it's current worth to people in different geographical locations. And in fact, Bitcoin is not the only asset whose value is been measured in fiat currencies, because even gold, silver and real estate. Because these assets all have their values been tagged to a specific amount of fiat currency, simply because fiat currencies is the only legal tender that are been known by the laws of a state as a payment medium that is been known for exchange of goods and services. Because it would have looked weird if Bitcoin was never tagged to any fiat currency, because everybody would have its exact market worth.

In fact, fiat currency is not just a piece of paper. Behind it stand the military power and law of state. Every country collects taxes from their citizen in their local fiat currency. So until we can pay income tax in Bitcoin, it is completely impossible for fiat currency to disappear or its need to end.
And the state will never give up its economic control or Monetary Sovereignty to a decentralized network.

Nowadays, I think Bitcoin is mainly successful as Store of Value. Just as we measure the price of land or gold in dollar, the same is happening with Bitcoin and may happen in future. Just as gold has not been able to become main unit of account for daily purchases despite maintaining its value for thousand of years, the future of Bitcoin may be exactly same.

However, if a major disaster occurs in the world, meaning if something like 3rd World War occur, only then can an environment be created for end of fiat currency.

The future has begun already because you can use Bitcoin in many applicable channels as you like; there are smaller fragments of Bitcoin you could achieve by having sats and mBTC.

You don't need to always convert your bitcoin to Fiat before you could make use of it, some platforms accept Bitcoin and you can also use it in payment directly, Fiat may exist together with bitcoin but not a must to exchange for Bitcoin, other unit of Bitcoin are what I have stated above and you can as well convert your Bitcoin to smaller units as you want.

Basically there are three functions of a currency: first is store of value, second is medium of exchange and third is unit of account. Bitcoin is strong in store of value but in unit of account it is still depending on USD (like we say 1 BTC = $65,000, not 1 BTC = 1 BTC). And this is only due to price volatility.

El Salvador is a real world example. They legalized Bitcoin but in reality most prices are still written in USD. Bitcoin is only used as a payment method.

In the long term, if the volatility of Bitcoin reduces, then theoretically it can become a unit of account. But this will take much longer time because at this stage, a price swing of 2 to 5% daily is normal.

No, Bitcoin has never really been used as a unit of account. Most transactions involving Bitcoin simply use it as a means of payment. Even Laszlo's pizza purchase was simply Bitcoin being used to pay for a pizza priced in USD, not a pizza priced in Bitcoin.

Even most signature campaign on the forum, while paying out in Bitcoin, calculate payment based on USD value. To this day, Bitcoin is still simply used as a means of payment in most cases.

The future is unpredictable, but personally, I don't think Btc will ever become a unit of account for the economy. Simply because government don't want that and will never let it happen.