I understand how the stardate system works in Star Trek and it doesn't really lend itself to casual conversation ("see you on star date 41645", as opposed to "see you on Monday") or recurring events such as birthdays ("My birthday is on 4X645" rather than "My birthday is on July the 5th"). How do they manage this in-universe?
In fact, do they even use relative terms such as "tomorrow" or "yesterday" ("Yesterday's Enterprise" of course, but that's out-of-universe). I can't remember any examples.
Yes, they use days of the week as well as months.
It's a running gag in Star Trek: Generations that the necessary equipment will be installed on Tuesday.
HARRIMAN: We don't have a tractor beam.
KIRK: You left Spacedock without a tractor beam?
HARRIMAN: It won't be installed until Tuesday. ...Ensign Sulu, try generating a subspace field around the ships. That might break them free.
(later)
CHEKOV: How big is your medical staff?
HARRIMAN: The medical staff ...doesn't arrive 'til Tuesday.
CHEKOV: (mutters in Russian) (to journalists) You and you and you. You've just become nurses. Let's go.
(later)
KIRK: Load torpedo bays, prepare to fire on my command.
DEMORA: Captain, ...we don't have any torpedoes.
KIRK: Don't tell me. ...Tuesday.
(Star Trek: Generations, Transcript here)
I briefly looked through other transcripts as well, and I am pretty confident that they are using the days of the week the same as we do. For example, this is from DS9: Crossfire:
KIRA: What is it?
ODO: I don't quite know how to say this, but I've given it a lot of thought, and er...
KIRA: Go on. Just say it.
ODO: I'm afraid I won't be able to make our Tuesday morning meetings anymore.
We can also frequently see them using months, for example, in DS9's "Whispers":
O'BRIEN: I mean, the way they were acting, they might've been trying to pull off one of those surprise parties that I can't stand, only my birthday's not until September, and believe me, as it turned out, I had nothing else to celebrate.
BASHIR: Ah, there he is now. I've been waiting for you, Chief.
There are also numerous other cases where they refer to days of the week or months.
They also use relative terms, for example in TNG's "Qpid": It starts out with:
Captain's log, stardate 44741.9. We have arrived at Tagus Three where the Enterprise is to serve for host of the Federation Archaeology Council's annual symposium. I look forward to giving tomorrow's keynote address with great anticipation.
Or here, in DS9's "Sanctuary":
KIRA: You wanted to see me, Commander?
SISKO: Actually, what I'd like to see is next month's duty roster.
KIRA: You'll have it on your desk first thing in the morning.
SISKO: That's what you said yesterday.
KIRA: I know, but I just spent the whole day yesterday...
SISKO: Talking to Minister Ro.
Worf's son Alexander born on the 43rd day of Maktag, Stardate 43205 (2366) TNG episode: "New Ground")
In the UFP and other known space travelling societies the old time systems of home planets and colony planets are still used. Earth people still use various Earth calendars with years, months, weeks and days to keep the date, and Earth people still use time keeping units like seconds, minutes, and hours.
So people who live on a colony planet or space station or moon base equivalent would use both the local time and date keeping methods to keep track of changes in local temperature and light, etc., and also the date and time keeping system of the home planet of their species to keep track of their ages and celebrate religious or secular holidays, etc., etc.
And no doubt they have simple computer clocks in their various gadgets to keep track of the time and date in both systems for whenever they need to know it.
And when people from various different species get together there are several different sets of date and time keeping systems involved. But converting between such systems should be no problem for the computerized time keeping systems they would have available.
And apparently stardates are used as an interspecies date keeping and time keeping method. Presumably people convert their units into stardates to communicate with aliens and when they are told stardates translate them into their units.
Memory Alpha says:
Stardate systems were used in certain cultures as early as the 2150s, when the United Earth government worked with Gregorian calendar dates. In 2154, Degra, a Xindi-Primate, sent a coded message to Enterprise NX-01 containing a stardate for when Enterprise should rendezvous with Degra's ship. T'Pol knew that it was three days in the future, indicating that Vulcans also had an understanding of stardates at that time. (ENT: "Damage")
The script for "Damage" says:
T'POL: I discovered a set of coordinates embedded in the document.
ARCHER: It's not a star system.
T'POL: There was another embedded number. A stardate, three days from now.
ARCHER: How far away are the coordinates?
T'POL: Four light years. We'll need at least warp three to make it in time. It's unlikely that Degra will wait beyond three days.
Since stardates are not used by Earth people in other Enterprise episodes, the stardate mentioned in this episode should be in a system used by various space traveling species in interspecies communication. This stardate system should have been around for years, decades, or centuries before the first episode of Enterprise, and might have been created by various non human societies before humans ever had warp drive.
If that stardate system was created by non humans with no input from humans, stardates would not be based on Earth time units. And if that stardate system was the system that came to be used by Starfleet in the era of TOS and the era of TNG, people who suppose that stardates are based on Earth time units would be wrong.
