I have many different saving buckets. As far as I'm concerned every expense is coming from "savings" it's just a matter of how often those expenses come up.
Some examples from small to large:
I pay most subscriptions yearly because it's cheaper, but I "charge" myself the amortized rate every month.
I put X/mo into tech, some months I might buy 2 or 3 cheap tech things (phone case, cables) whereas other months it just all gets saved for something big like a PC upgrade.
I keep a few grand as an emergency fund, I save X/mo until I have Y, then just leave it unless I dip into it.
I have a repair budget, I save X amount every month, if It ever reaches X*20, I split it in half and keep half as repair budget and put the other half towards a big renovation/upgrade.
I have a mortgage budget. I am fortunate enough to afford more than my minimum payment. Until my mortgage comes up for renewal It is more efficient for me to invest that cash and then dump it in when the renewal comes up. Assuming interest rates don't go too much higher than now I think I can keep my upcoming minimum to be the same as my current minimum + my current mortgage savings.
It's a line item in my monthly budget. I run some future simulations to ensure that number is set high enough to at least be in the ballpark to achieve my future goals.
I treat the monthly number less like a goal and more like a required expense to attain what I am hoping to attain.
I've never thought of savings in terms of setting a target amount per month. Instead, I've always targetted my spending so that it is at most 70% (though often this has been lower) of whatever cash I regularly have coming in. Any income I get above my regular pay is 100% sent to savings.
I budget assuming 2 paycheques per month, though I am paid bi-weekly. Off each paycheque I automatically withdraw a fixed amount to my shared household spending account and my personal one. All of this spending is around 65% of each paycheque (after all deductions). It includes everything from necessary life expenses, fun money, and travel to emergencies.
Everything else is saved. So, to be more precise my savings come from 4 sources:
Automatic paycheque deductions for DC pension and employee share plan
Everything that is leftover after the automatic deductions for spending
2 'extra paycheques' each year (24 pay periods in spending budget vs 26 actual)