UK here. Terminology may differ, I think what is being discussed is a
notice period. Normally a resignation is either accepted or rejected pretty much on the spot and if accepted is officially seen as a more amicable departure.
Handing in your notice to leave is something the employer cannot reject. Depending on circumstances you might try to resign first. Sometimes you may be "asked to resign"[0] and made a nice offer to encourage you to do so, offer will almost always have an NDA attached to it.
[0] this can be preferable to the company as it can be quieter than sacking you
> This guy(gal?) is really required to submit 3 months notice or face some sort of repercussions (financial, I assume)?
Yes. The worst they can do is take you to court for breach of contract though. That can be expensive and embarrassing for both parties so unless they are whiter than white[1] or bitter enough companies often don't bother.
Potentially a big risk to take though. And in an industry that is hardly free from nepotism, you probably don't want the reputation of being someone who walked that way.
Three months is most common in skilled jobs, one month otherwise. In retail work and other low-skilled professions two weeks is not uncommon, though here people often do just walk away as the hassle of chasing them for breach of contract is massively not worth it.
[1] i.e. there is nothing you can say on public record that looks bad for them going forward
> Does this 3 month period apply in reverse (employer can't fire employees without a full 3 month warning)?
Yes. What usually happens if they want to have you gone without notice and don't want to make you an offer to resign "amicably" is that they put you on gardening leave. You are still employed and paid for the notice period but are told to stay home. They can't stop you looking for new work[2] as long as you don't do it "on their time"[3] but you can't start new work without resigning and them accepting that resignation until the end of notice period without the same risks as if you had just walked out without notice yourself[4].
Gardening leave can also happen when you hand in your notice, though if you want to leave and the company wants you to leave then they'll just agree to waive the notice period.
[2] Some contracts have clauses that say otherwise, those clauses are not legally enforcible. Some even have clauses saying you have to actively look for new work, these too are not legally enforceable.
[3] Which is impossible for them to prove unless perhaps they have you check-in regularly while you garden.
[4] Though they are likely accept the resignation as at that point they can stop paying you for the notice period, unless you are going to work for a competitor, or they otherwise feel like being dicks about it.