We have savings but they're not in specific pots and they will be spent as an when needs etc are identified.
We are fortunate that we have enough of a surplus that we'd just buy a new cooker, washing machine, pay for car repairs etc - it all goes on the credit card that's paid off in full every month so there's some mitigation of the ups and downs before savings have to be touched. If you couldn't go this, you could dip into your general savings pot for something like a new cooker?
But you're right that you need savings for things like car repairs, christmas, holidays, insurances, loss of income, etc etc. Ideally you should also be in a position where you can pay for these things and things like car replacement without having to borrow money, because that increases your essential commitments which makes it harder if you lose your job or become ill.
Reading between the lines, one of you wants to buy something non essential and can see money sitting there waiting to be spent, and the other one wants to keep the savings topped up?
Getting the balance right about this can be difficult and depends on a lot of variables such as pension provision, income/job security, whether you have DC and need to save for university, whether you can cover essentials easily and all this before you start thinking about how you split your surplus amongst all the competing non essentials.