Random guessing. None of this is canon, just me spitballing concepts.
1. Because Sailor Venus' capacity as a body double for Sailor Moon (as seen in that one episode with Kaolinite) worked so well it fooled the cats, and Artemis legitimately thought she was the moon princess.
2. Because Artemis is a dork, and Luna was slow.
3. Artemis has some unknown special connection ( :love: ?!) with Sailor Venus from the Silver Millennium era.
4. Minako is an early bloomer on magical energy, so the others weren't ready to awaken quite yet.
5. They found and considered Usagi first offscreen, but decided she couldn't possibly be the right one, and looked elsewhere first before resorting to Usagi after a trial run of Sailor Venus.
6. Because Minako is secretly the real moon princess and nobody figured it out by the end of Sailor Stars.
7. Because Minako is not actually the reincarnation of Sailor Venus, but rather an imposter mole / undercover sleeper agent from the Dark Kingdom waiting for the perfect moment.
8. Because Minako is Sailor Cosmos Chibi Chibi.
As for why her costume is different...
1. Maybe Minako's powers awakened imperfectly, and the Sailor V form is a weird janky didn't-quite-take kind of form?
2. Minako has her own rarely-seen 'Super' form in the first place, which allows her to melt youmas by kicking them in the face like we saw in London, but this 'super' V form is deactivated for some reason by the time she joins the team. (The mask is a special upgrade much like Usagi's wings are a special upgrade.)
3. Artemis goofed up the settings on Minako's transformation pen and only fixed it later.
4. Artemis damaged Minako's transformation pen and only fixed it later.
5. Because Taku stole Minako's transformation pen from Artemis and messed with the settings to design a sexier outfit.
6. Because Minako is a secret Dark Kingdom agent, using the transformation pen has different results, which she was able to hide later before Artemis was any the wiser.
7. Because everyone originally would've had a Sailor V type form, but the transformation pens went through some kind of automatic upgrade when Usagi was first awakened, which sent them all to the next form level before they ever saw those V-type forms.
8. Because Minako had a powerful subconscious desire to look awesome.