I mean, days of the year is fine as is because that matches seasonal cycles which is important for agriculture and the like. Ofcourse along the equator lunar calenders were historically used because they don't have different seasons but a calender used by everyone keeps things consistent for trade and the agricultural trade is still one of the most important things for humanity so it best remains a solar calendar.
As for time, 60 and 24 and in general multiples of 12 are great units because they can be easily divisible in many ways. This is why they were established as such in the first place, because Babylon had a big thing for what is called "extremely composite numbers" which are basically just a number that has more factors than any previous and equal to all subsequent ones (until the next extremely composite number) and all multiples of 12 (as far as I know) fall into this category. This means that units that are multiples of 12 both easily divide into each other (making it useful for changing between units) and easily divide into many other numbers making it easier to schedule a day as you have nice clean boundaries.
Basically it comes from the roman calendar. Which is based on the sun and the moon cycles.
A year is a full turn around the sun. It takes 365,25 days. So 365 days, and 1 more once every four years.
A month is based on the moon revolution around the earth, which is 29,5 days.
But in the past they weren't as good to measure these times, so they used more approximative calendars that they would fix by adding days to synchronise the calendar on the moon or year. Roman for example had the month always start on a full moon.
For the months, you had then some emperors who wanted their month to get more days. I think the 7 days week comes from Christians, the bible talk about it. 7 days allows 4 weeks in one moon cycle. Roman used 10 days weeks, so 3 weeks in a moon cycle. Both are inaccurate, so there were days added or removed to synchronise regularly.
Julius Caesar basically made the calendar we have now because the old one was a mess. Then, in the XVIth century we got the gregorian calendar, which is almost the same but fix some problem the old one had.