This article doesn't exactly describe common ATX power supplies and doesn't answer OP's question about shared inductor.
One common type of old group-regulated PSU is found on page 6 here:
https://www.onsemi.com/pub/Collateral/TND350-D.PDFThis is combined with a forward topology primary. The secondary windings of the transformer output two square waves in approximately 12:5 proportion but higher than 12V/5V, the positive halves are applied through diodes to two inductors in 12:5 proportions wound on a common core which implements a sort of dual buck converter. Output voltage is regulated by varying duty cycle like in any buck, but the two voltages are locked together in approximately 12:5 ratio.
Another option found in low power or really cheap PSUs is flyback topology. Then, pulses from the secondary directly charge output capacitors through diodes and the dual buck coil is not present.
The 3.3V rail is generally derived from 5V, either by means of a mag amp buck converter (described in both PDFs linked here) which requires two coils (the mag amp coil and an ordinary buck coil) or by a switcher (the mag amp is replaced with a transistor) or by linear regulation (no mag amp, no buck coil, only one transistor and terrible efficiency). This choice is independent of how the 12V/5V rails are generated, although there tends to be a correlation: the cheapest PSUs use flyback and linear 3.3V rail, the better use forward and buck/mag amp.
So the number of coils can be anywhere from zero to three and these are all group regulated designs. Sometimes much smaller inductors are used for ripple filtering, placed inbetween two capacitors.
To drastically oversimplify it, modern lower power PC's don't really draw anywhere as much +12V as older PC's did, and as time goes on +12v is even less needed however +3.3v much more. Lots of components that historically use 12v + 5v together now only use 3.3v, and in the last decade largely only 5v and no 12v. A circuit that needs a certain level +12V load to be stable is not a good idea if you barely sip +12V and need really accurate and large amount of 3.3V and 5V.
It's the exact opposite. Computer chips used to run on 3.3V and 5V straight from the PSU, but now they use much lower voltages which are generated on the motherboard by small bucks drawing power from 12V. Some chips may still use 3.3V for I/O interfaces specified at this level, and the largest remaining consumer of 5V is USB and disks.