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?