The mechanism is the same, if a school reports unacceptable levels of unauthorised absences to a body which can issue the penalty notices, the parents can be fined. This was introduced in the 2003 Anti-social Behaviour Act which does not specify it applies to state schools only.
The (1944?) duty on parents to provide a suitable education by regular attendance at school or otherwise also applies to all parents .
The mechanisms exist. Fines (indeed even prosecutions) can be used on any parent with a child enrolled in school who does not then attend it.
Authorising attendance is at a head teacher's discretion in both sectors.
In independent schools, you may well find that parents are just asked to remove their DC if attendance is awry, rather than going down the route of enforcement notices. That option isn't available to state schools.
Also, the economics of it are different - if you're paying a few hundred pounds a week for your DC to be in school then you'd probably want them to be there. If you can afford to pay for that yourself, then you may also be less fussed about securing discounted holidays during the off-peak.
And good attendance is a regular condition of bursaries, and scholarships so parents whose DC have those awards wouldn't dream of it.