I took bass guitar lessons years ago, but decided I was going to start self-teaching again. I have 2 basses, a Squire P-Bass, and a Fender fretless J-Bass. I was thinking about teaching myself to play on the fretless, as it would be less forgiving, and teach me exact fingerings. Am I completely insane? Should I just start on the P-Bass to learn some technique, then move over to the J-Bass after?
Welcome to the wonderful world of non-standard intonation. You will like it here.
I have answered here on instruments where you have to dictate the intonation: slide and steel guitar and theremin. I haven't mentioned violin/fiddle, because by the time you're good enough for it to be worthwhile to ask questions in this sort of forum, you've already learned more about different intonation schemes than most of us will ever know.
The first thing is, if you're coming from a standard guitar/bass world, your idea of intonation is something you set with a screwdriver and truss-rod wrench when you change string sizes. Once your electronic tuner says "OK", you proceed with all assurance with the knowledge that the third fret on the E string is a G, and if it isn't, it is now. That's fine, but you're going back to basics and your ear recognizes out-of-tune mostly in context of notes that are in-tune. So, first and foremost, practice with tuned accompaniment. Other players. Tracks. MIDI files. The radio. You have to teach your ears to recognize what in-tune is as you teach your hands to go there.
I would also seriously think about styles and genres. Of all the electric fretless bass players I can think of, they're all jazz guys. OK, there's one rock fretless guy, Tony Franklin of the Firm, etc., but there's also one electric metal violinist, one classic rock flautist, etc. When there's one, there's not two. If you want to play jazz bass, great. If you want to be the signature fretless thrash bassist, cool. Just know that tools have purposes.
The important thing is, your first goal is to learn to play in tune on the thing, by ear, not by eye. All else can come after.
You're not insane, but you do run the risk of becoming so frustrated that you end up giving up playing, which would be a shame. Obviously, people can and do learn how to play on fretless instruments---upright bass, cello, violin, etc.---and do just fine. But it takes a lot of discipline, time, and patience to get to the point where you can play well.
This is all assuming, of course, that you have musical (as opposed to just technical) reasons for wanting to learn to play a fretless. If you're going to put the time and effort into learning on a fretless but end up playing the fretted bass in actual musical situations, then don't bother with the fretless in the first place.
If you have both kinds already, you should use them both! Some techniques are almost impossible on fretless, so it might be a bad idea to use it exclusively. But most ordinary stuff works quite well, so if you'd like to be able to play fretless at all I'd recommend practising as much on it as possible: like you said, it will improve your intonation skills, and it doesn't have any intrinsical drawback over practising the same stuff on fretted. I, for one, practise much more on fretless, even though I play mostly fretted in my bands. With such a policy, you get better on both instruments, but – while the intonation on fretless is not acceptable yet – still don't annoy the public with bad notes.
However, while there is no intrinsical drawback, there may still be reasons to rather practise some things on fretted even though they're possible on fretless. For instance, fretted basses are often set up with higher action / heavier/harder strings then fretlesses, so practising on the latter might not sufficiently exercise your left hand's power. For similar reasons, it may fail to keep the callusses on your right hand strong enough to survive a gig on fretted. So, once again, you should practise on both instruments.
As to "completely insane": definitely not. It's not trivial to play correct pitches on a fretless, but it's very doable. If your bass has lines, you might even play completely by eye rather than by ear (though I'd strongly recommend avoiding that). If you are slightly off, it sounds much less annoying than e.g. an out-of-tune vocal note; it has to be more than a millimetre on the fretboard to sound really wrong.