If it's v6.8 of your internal software product, no, the deadline doesn't usually matter that much. Maybe you are a bit behind your competition, but the only problem with just being at the 6.7 feature set a little longer is a small percentage of customers not sticking around. Your manager probably overpromised and underestimated, but aside from a bit of abuse and panic, you're right, it doesn't really matter.
But if you're one of 120 companies building parts for the 2025 F150 and your part isn't ready and the other 119 companies are ready, whether it's your supplier or you or your customer who is behind, you'd better believe that they've got you over a barrel in the contract. Like, the contract has a line that reads "for every minute of downtime after [date] you owe $25k". Your PM hopefully underpromised with an aim to overdeliver, but missed really bad. And all the checks and balances failed. If you've got adequate financials you can survive a miss once in a great while, but two or three in a row and even big shops will go under.
It's often important to understand the consequences of missed deadlines that your coworkers, vendors, or especially customers are accustomed to, people have wildly disparate experiences in this area.
Or I have a product that relies on certain hardware. I've already bought/leased that hardware, so I'm now paying depreciation/lease payments for that hardware. Every week I delay the launch I'm losing money on those payments because the hardware is sitting unused.
If your product isn't ready and your competitor's is, you lose contracts and possibly life time customers.
Modernizing and moving functionality to the cloud now instead of in a few months prevents renewal of a expensive contract.