Tone is the whole signal chain - your fingers (or a pick), strings, pickups, wood and the bass itself, amp choice, etc. How you play influences, of course, but the HH Stingray isn't going to really sound like a P exactly. However, Geddy still sounded like Geddy on the R40 tour no matter what bass he played but each bass had a different "sound".
So the argument really is around "your sound". You need to understand what that is in your head and you can dial it in whether it is a $300 instrument or a $30,000 one. I've done that on other people's basses and amps.
The small differences - fit/finish, playability, neck shape, etc., all matter too. No one is saying that a $300 instrument is bad. Today's $300 instruments are far superior in most cases to those when I was growing up BUT to say someone should just play a Squier/Epiphone/whatever and call it a day is ridiculous. We all have reasons for liking what we like. Some instruments help us get there, others won't.
Play what you can afford and makes you happy. If all you ever want is a Squier and are happy, cool. If you aspire to own something else? Awesome. J Wilson's post is spot on.
So to me, I think it's in your head. Now, for sure, it's a better-sounding 'me' on Alembic than $5 Squier pickups, worlds apart construction-wise for sure, but the Squier JW ain't bad considering the cost differential.