Fender owns Squire so it will be essentially the same guitar but made from cheaper wood and cheaper components.
Back in the 80s I bought a Charvel (Japanese) as I couldn't afford a Jackson (US made) - the cheaper Floyd Rose tremolo was disappointing but it was fine to play. Eventually switched to an Ibanez RG 570 which was super nice to play.
If "essentially the same" means "subcontracted for the company that owns the name and copyright to headstock shape". Original Fenders have already changed so much that the company sells replicas of their old products, and on the other hand there's nothing that prevents competitors (including G+L, the company Leo Fender himself started after selling the more famous one) making the exact same guitar if they change the shape just enough that it can be recognized as different brand.
Essentially the same in that an electric guitar is a few pieces of wood with some hardware and electronics. How well those pieces are put together and the quality and specs of each component makes the difference.
The type and grade of wood used will also affect the string vibration and what overtone frequencies are refected back into the strings. It takes 20-40 years to mature a mahogany tree so while your boutique makers will spend the time to source the perfect timber you won't get that for a mass produced guitar.
There are guitarists who swear by their guitar tech who will adjust everything from bridge height, neck tension, even reseat and glue the neck before they pick it up the first time. I pulled the neck off my old Charvel more than once to get a lower, more even string height along the fretboard. Second time I filed back a few frets that had nicks or weren't quite evenly set and sanded the epoxy coating off the neck and replaced with shellac for a smoother raw wood feel.
Ok, I misunderstood that you meant there's something special in Squier that would make it more Fender-like than a G+L or any of the almost-clones that don't have any direct relation to the original manufacturers.
Back in the 80s I bought a Charvel (Japanese) as I couldn't afford a Jackson (US made) - the cheaper Floyd Rose tremolo was disappointing but it was fine to play. Eventually switched to an Ibanez RG 570 which was super nice to play.