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Yes, emergency funds are supposed to be in fiat currency, but if someone chooses to save it in stablecoins, I think it's not a big problem. Saving in stablecoins can work too.
You need to think about how liquid your emergency funds are, and consider the various kinds of emergencies that might come.
Sure it may well need to be the case that you have around of a month of expenses that are always in your local fiat and easy to get to to pay all of your monthly bills in the event that your income dries up and/or you suffer from some immediate additional expenses.
The less liquid the fund, then the more chances that you are taking, and yeah, maybe it takes you 1-2 weeks to get at such funds and then to cash them out in the form that you need, and that might not be problematic, yet if you have some uncertainty about the liquidity of some funds and how you could have trouble getting access to them for several weeks, then you likely would have redundant systems of funds in other places to make up for potential problems of funds that might have issues, including issues with the custodian if you are not holding them yourself and able to sell them directly yourself.
The short experience gained in studying this thread suggests that every investor needs an emergency fund to maintain their family for at least six months and to deal with unexpected situations.
Emergency funds is not the money for maintaining your family but it should be used to solve a real emergency that happens which if not taken care of, it will lead to more destruction and loss in your life. It's only when you have lost your job that you can use your emergency funds for survival until you get a new source of income so that you don't touch your bitcoin investment at all.
Of course, when we are in the earliest stages of building our bitcoin investment and perhaps also building up our emergency funds at the same time, it may well end up being the case that we end up tapping into our emergency funds for non-emergencies because we are still in the earliest of stages of building up such fund, and surely we are also running risks while we are building up our emergency funds, yet at the same time our bitcoin stash is likely not going to be much BIGGER than our emergency funds, at least going from the amount of value that we put into each of them.
Ideally, once we get the emergency fund up to at least three months of expenses, and perhaps our bitcoin investment is also getting to be at least that size too, then we may well strt to build up various other kinds of reserve funds, so by then we would tap into our other reserve funds prior to touching our emergency funds, and our emergency funds would be the last thing that we touch prior to having to dip into our bitcoin, so we already realize if our finances have gotten to such a stressed point that we are having to dip into our emergency funds, then we should realize that we are in a place that we don't want to be..
and that ideally, like you said Merit.s, we are not going to be touching our emergency funds at all absent an actual emergency and not merely our own inadequate cashflow management skills... but it can take a while to build our finances to such strength that we both have various kinds of back up funds beyond just our emergency fund, and we also are attempting to invest somewhat aggressively into bitcoin too, and perhaps our bitcoin investment is growing faster than our emergency funds and our back up funds, since once we get our various back up funds to certain levels, we may well not need to keep growing them and we likely are going to want to focus more on aggressively (perhaps?) building up our bitcoin investment.