My bass teacher told me that "Every true bassist needs a P Bass."
Agree or disagree?
This is edging to "no true Scotsman" territory, where your teacher is attempting to define "bassist" in part as "owning a P-bass". If that's your teacher's intended implication here, I disagree.
P-basses are indeed very common and very well-liked instruments; they are a "one-trick pony", one of the simplest electric instruments you're likely to encounter, but that trick and that simplicity are the true power of the design. The bass just
works, across practically any genre that uses electric bass guitar, and it's so common in the music industry that it really
has become a definition of "recording engineer" to know how to get what you want for the track out of a P-bass.
All that being said, it's not
essential to have a Fender Precision Bass in your collection/stable/harem. I answered "Yes, but you can get a P/J", because that's been my experience. My first bass was a Yamaha RBX-260, which
happened to have a single P pickup, but you wouldn't actually call it a Precision bass, and I haven't owned a one-pickup, two-knob P configuration since I started playing in front of other people. I made a sideways-ish move when I started gigging in the bar circuit, from my Yamaha TRB toaster-top that had been my primary bass for a decade in church (and still has a prominent place in the harem) over to a Sire P7. I did so specifically to have access to a straight-up passive P-bass tone, and a more conventional Fender look, while not losing bridge pickup bite and the flexibility of active electronics (not to mention that B-string, which has become integral to my playing). That bass has been my primary gig weapon ever since, and my newest bass (Charvel Pro-Mod San Dimas PJV) is also a P/J, bought to look a bit more rock/metal than the Sire's very traditional lines while again not being so far removed from the basic Fender P recipe that sound guys don't know what to do with it.
In short, it's less of an "every true bassist has one", and more a "you don't have to be crazy to work here, but it helps" mentality. Having access to that sound is always a plus especially if you plan on laying down studio tracks, but it's not a hard and fast requirement of getting/keeping your Bassist Card to have a Fender Precision to hand.