I know that I've heard of this type of rule in retail/restaurant-type workplaces, though I believe it's typically only in the case of "calling in sick", not "taking a vacation". Even then, though, it can be a difficult rule to work with, if many employees are unwilling or unable to fill in on their day off.
That said, employers can put in place many different rules specific to their workplace, and you will need to abide by them if you wish to work there.
If you want to change your manager's opinion, your best bet is to speak with him/her and ask why the new policy was put into place. You can also ask what to do if you can't find someone to cover your shift. "If I get the flu next week, and I call everyone who's not scheduled, but no one answers the phone, what should I do? Should I come to work and expose everyone else? Should I call in and explain 'I can't come in, but no one else would take my shift'?"
If they can't answer this (or if they answer it in a way that you find unsatisfactory), but they continue to insist on the new rule, then you need to accept that this is the way the company wants to work going forward, figure out how likely it is that you'll run afoul of the new rule (do you get sick often? are your coworkers likely to ignore their phones when they're not working, so that in an emergency, you're unlikely to find someone to take your shift?), and determine whether you want to continue with this job or begin a search for a new position.