What do phase dots on an inductor mean?

The dots just indicate the polarity of the windings on the schematic.

Currents entering on the dotted ends of the windings will produce magnetic flux in the same direction, whereas if you have current entering one dotted end, and leaving another dotted end, the currents will produce opposing flux.

If you're looking at something like a power supply schematic, the dots show you the ends of the coils that have the same phase angle.

If we're thinking about same thing, then dots show relation between coils. Here's an article about that.

If I get the dot relation correctly (and I'm not sure that I do), if current is going into the dot on one side of the inductor and the other dot is on the same side of the inductor, that means that current is going into the dot on the other side too.

How important it is to have dots in the same place is up to design of the circuit. In some cases it may not be relevant, but often it is important to take it into account.

Normally we apply AC voltage on the (input) primary coil of the transformer and we have an (approximately) resistive load on each (output) secondary coil of the transformer.

In that case, the dotted end of every coil will reach the positive peak voltage relative to the non-dotted end of that coil at (approximately) the same time.

Also at that same instant current will be flowing into the dotted end of the primary coil and current will be flowing out of the dotted end of each secondary coil.

The dots describe which way each coil was wound. If I take a CAT5 cable and wind it around a ferrite core, each wire-end at one end of the CAT5 cable is dotted. Each wire-end at the other end of the cable is not-dotted.

I honestly don't know if dot orientation makes any difference for a SEPIC converter.

I know that the dots are important in the very similar "Coupled Inductor Cuk Converter" and "Integrated Magnetics Cuk Converter" ( The Four Topologies ). If one accidentally swaps the two ends of a coil in that converter, you get increased (worse) ripple on the input or output or both.