How important are the seasons going to be for people who have to spend the vast majority of their time indoors due to the lack of a breathable atmosphere – and possibly even underground for protection against radiation?

Maybe one day Mars will be terraformed and given a breathable atmosphere, and maybe even some kind of artificial magnetosphere can be created to protect against radiation. But that is going to take many centuries.

The fact is that modern societies don't really need a calendar synchronised to the seasons. Most jobs nowadays the seasons are not very important. This is especially true in parts of the world where the difference in weather between the seasons is less marked (like where I live, here it never snows in winter). Fields such as agriculture where seasons are important can always keep track of the seasons independently from the calendar, or using a special purpose agricultural calendar. In fact, that's what is traditionally done in much of the Islamic world – the Islamic calendar was used for most purposes, which is out of sync with the seasons. Those jobs for which seasons were important, such as farmers, also used a separate solar calendar. It was only in the 20th century that the need to participate in the global economy created pressure to use the Gregorian calendar instead.

> I like Kim Stanley Robinson's approach, as described in the Mars trilogy. They use standard seconds, but at 12:00 midnight, the clocks just stop for about 39 minutes

I don't know why you'd do that. On earth, clocks run from 00:00:00 to 23:59:59 each day. On Mars, you'd just make them run to 24:39:34 instead. That way you can actually get precise times for events that occur in those extra 39 minutes and 35 seconds that happen each day. (Also, I'd hope people on Mars would be using 24 hour time–I'm one of those people who changes all their clocks to 24 hour time on Earth)

> Martian colonies will still care about Earth calendars, but only to the extent that they have to deal with Earth

For the first few decades at the very least, Martian colonies will be dealing with Earth an awful lot. A lot of goods will have to be imported from Earth. There will be a lot of focus on what they can locally manufacture on Mars, but they'll start with basic stuff like foodstuffs and simpler manufactured goods, and it will be a long time before Mars has leading-edge semiconductor fabs or a lot of other stuff like that.

There will also be continual immigration from Earth, and also people returning to Earth (whether temporarily or permanently). That's going to psychologically link them with Earth.

Martians are also going to consume a huge amount of cultural imports from Earth. You think on Mars they won't have Netflix, YouTube, etc? They'll replicate the content library into a Mars-based CDN and people will be watching it on Mars. It is going to take a long time before natively produced Martian media/entertainment content becomes more popular than the Earth-imported stuff. And I think that's going to tie Martians to Earth at least as much as trade ties and movement of people back and forth will. Martian colonies will undoubtedly develop some unique cultural aspects, but a huge amount of their culture is just going to be Earth imports.