We also don't have any particular reason to think that it will

Also, on Earth we already have situations near the equator where there's not really a significant change in the weather from one season to the next, or near the poles where for parts of the year days are considerably longer/shorter than elsewhere on the planet, we also have people living in scorching deserts and frozen tundras, at high altitudes with thinner atmosphere, etc. and despite all that variation we don't really see major differences in how quickly children mature.

The differences could be even more profound on other planets of course.

There have also been studies where people have lived in caves or bunkers without natural light, clocks, or other cues about the time or day/night cycle, and it's been found that we stay pretty close to a 24 hour circadian rhythm (usually slightly longer actually, but within a few hours of that target,) so it seems like that's something that might be hard-coded into us. Of course those studies have been done on adults who have had decades to acclimate to a 24 hour cycle, so it's plausible that kids raised in a different environment would naturally adapt to a different cycle, but since we're probably not going to be sending unaccompanied minors to the stars, those same kids would probably be raised by adults who are used to a 24 hour schedule and would raise those children in the same schedule.

You might see some divergence from that over the years and multiple generations, but if there's a 24 hour clock present, and people decide to stick to that, I suspect that would work just fine. It would probably come down to whether it's more beneficial for people to be in sync with the rest of humanity, or to be on the local cycle. My money's on the former, since we probably aren't going to need to worry about hunting for sustenance or avoiding predators, or other such things that our circadian rhythms evolved for.

Something we can't really account for though is if different gravity would affect how quickly children mature. It will almost certainly have an effect on how they mature with differences in height and muscle/bone density, but I don't think we can really say if it will change how quickly their brains develop, when they begin puberty, etc.

There's other factors that could play a part as well of course, the composition of the atmosphere, the intensity of radiation from the star you're orbiting, diet, exercise, different mutations that could arise over the generations.

It's not Earth-focused, but Eve Online had a minor subplot about how the coordination of time between multiple competing human civilizations became a major political problem because no one wanted to accept the system promoted by a rival empire.

I think that was in The Moon is a Harsh Mistress, iirc

It's the symbolism. Of taking authority over their own lives back from the central authority.

Doesn't really mean much in a practical sense. It's more about making the people feel more empowered.

I recently read an interesting book called "the mimicking of known successes" which is a mystery/romance novel but the scifi world building is the part I most found interesting. Humanity had colonized Jupiter, but ultimately died out on earth due to complete ecological collapse. Anyways on the timekeeping subject they didn't really talk much about time, but it was noted that inhabitants experience 2 day/night cycles per regular earthly circadian cycle, and people will generally end up on sleeping for 1 day/night cycles and being awake for 1 day/night cycle and people can be on different cycles entirely, so presumably time is tracked within a cycle and people just sleep through every other cycle

it doesn’t do daylight savings.

And that's a blessing.

Seasons still play a part. While Green Day would have to sort out the cadence for the rewrite, referring to fall would make sense instead.

Months are useful on Earth

Are they? I've found them to be the least useful division of time. They don't line up with the days of the week, or the phases of the moon, or the seasons, and you need a mnemonic to remember which ones have 28, 30, or 31 days. Now if we had 13 months, each with 28 days, that could be a bit more useful...

We’d perhaps have to define it as the centre of the planet crossing the unique plane that both passes through the star and the centre of the galaxy and is perpendicular to the planet’s orbit relative to the star.

I’d rather define it as the point where a ray from the star through the planet crosses the pole passing through the galactic center perpendicular to the plane of the galaxy. Make it depend as little as possible on the planet’s orbital characteristics. (And yeah—use barycenters for binary stars and planets.)