You've only just started working. The cost of living will eventually go up for a myriad of reasons as you go.
If you're referring to savings as cash in the bank for a rainy day. I barely have a few grand available. I think for myself and many others with good credit and responsible usage, the ease of access to credit kind of gives some extra headroom should we ever be in trouble. At least to last us until the problem has been sorted out.
Moreover, I'd only keep at most, 10% of my worth in cash. Any uninvested cash provides no returns and is at the mercy of inflation. Therefore I keep my ISA maxed out, equity locked into property and a few index funds for steady growth.
As for people not investing their money, it's simple. They spend more than they earn or too close to that point.
I consider myself quite a heavy spender. In the past 3 months I have spent £600 on new phone, £4000 engagement ring, £4000 on holidays, £75/mo on cable, £300/mo on vehicle upkeep, £100 a week on fuel, £700 monitor for PC, £1000 in gifts, £500 on vehicle tax, £700 vehicle warranty, £210 new tyre, £2500 private license plate, £18000 paying back debts and an absurd amount on a new car. Oh and I regularly eat out and rarely stay in to cook. There's many more things but that's what I can think off the top of my head. A lot of those purchases are occasional, such as holiday (haven't had one in 2 years), replacing a 2year old phone, new car purchase and of course, engagement.