The winner: The Squier is IMO the better bass in every category. It’s got the old school vibe and tone which really shine through.
I agree the Squier VC is a very well made, beautiful baass. I compared the early Squier VC Mustang I owned for about a month with my Fender Mustang LE PJ and I found the Squier sounded somewhat weak compared to the P pickup of the PJ. Adjusting the pickup height and turning the amp up for the Squier and EQ didn't improve things much. I went to return the Squier and found the shop had a minty very lightly used first version Mustang PJ with the rosewood fingerboard and it too sounded noticeably better than the Squier so I returned the Squier and bought the PJ for a bit more than I paid for the Squier.
Your description of the tone of your Squiier makes me wonder if the pickup I had was defective or if they improved them.