First, if you have a prearrangement with the bank to provide you overdraft loans, then it's just a loan. It sounds like that's the arrangement you have, because I've never heard of a bank tolerating plain overdrafts every month.
Now just to disclaim this because it's important for others -- Without such an arrangement, overdraft is serious business. Now mind you, countries vary. But in the USA, where I sense OP is from, it needs to be covered right away, or it will be inferred that you intentionally wrote a bad check / did a charge you knew you didn't have the money to pay. That has consequences beyond fees. Least, they can close your account and place you in the ChexSystems blacklist -- and you will not be opening a bank account for 7 years. Lots of people become "unbanked" that way, which is a real problem if you're trying to create an emergency fund!
That said, I disagree with my colleague that emergency funds are for people with budgets who already are living within their means. That would be to say that financial education is only for the already financially educated. Not at all.
Emergency funds are one way that you learn to live within your means. See any of the people who talk about paying yourself first. Taking a chunk of your paycheck and diverting it straightaway into emergency fund is a great way to avoid spending it some other way.
It's not like you spend the whole month hoping you have enough money at the end of the month, hoping you don't go to the bar too many times, or hoping you cancel the $150/month Comcast plan. You do those things because you have to, because you ran out of money, because you already diverted the money into the emergency fund at the top of the month.
The reason to save for emergency fund first and not pay off debt first is the credit line won't be there for you when you need it. at the first sniff of you being in trouble, banks are quite aggressive about dropping $20,000 lines down to $500 credit limit for no reason, or dropping $29,000 lines with $11,000 balance to $11,200 credit limit. And don't be surprised when all your lines do it in the same month.
And, this can happen simply because the country is in a recession. The Great Recession saw a whole bunch of people's "credit card emergency fund" get erased before their eyes. I was listening to Suze Orman around that time, and it was an endless parade of tears.