There’s no real canon explanation for why it’s there, but I can certainly share my thoughts! First, I think it’s worth noting that we only see the name there after the curse drove Isaac mad. Being a devil forgemaster, he’d easily be able to remove/repair the sole too, so since it’s still there it definitely has personal meaning to him.
A name written on the sole of your shoe can indeed mean you “belong” to that person (a la Toy Story, where Woody, Buzz etc have Andy’s name on their feet). That’s certainly fitting for Isaac, knowing he and Hector’s delicate relationship. In some ways, Hector owns Isaac without even realizing it; his favor with Dracula and his proficiency in combat and demon magic kindle in Isaac an obsessive need to train harder and more deeply immerse himself in Dracula’s magic. He needed to maintain pace with or surpass Hector if he wanted to stay in Dracula’s good graces--or to simply survive. It’s kill or be killed in Castlevania, especially as a general, and I doubt Isaac trusted Hector to stay his blade if they fought. At his darkest, I’m not sure Hector would have in the first place. So, yeah. Hector had immense power over him even before the curse twisted Isaac’s mind.
Writing someone’s name on the bottom of your shoe can also be part of a ritual to control or dominate them. Noting their rivalry and Isaac’s eventual madness, that’s also quite plausible. Both the boys were up to their ears in black magic and Isaac was hellbent on revenge, after all. Honestly, this, combined with the explanation above, is my headcanon for it. I think it’s completely reasonable to assume Isaac cast hexes on Hector before deciding to frame Rosaly. None of them worked because Hec was hiding out on holy ground, but the name remains, if only to remind Isaac that he has unfinished business with him. That he can neither forgive, nor forget him or his betrayal.
Some old legends mention writing the name of a lover on the bottom of your shoe, too. I’m not well versed on this one because it’s never really worked with the rest of my headcanons and I think it’s silly. I mean, why would you want to step on the name of your lover everywhere you go? But given some of my headcanons on Isaac’s overall character--which is a whole other post haha--it might not be a stretch. I’ve seen some argue that jealousy was an additional factor in Isaac wanting to kill Rosaly, too. Isaac’s red hair alone would’ve historically marked him as evil, so any kindness or even neutrality on Hector’s part could’ve seemed extraordinary to him, or even misunderstood as attraction. Maybe he loves Hec or is just obsessed with him, be it in the romantic or sexual sense.
I hope that helps! Icould go a little more in depth with personal headcanons, but I’ve got a cold and need some soup.