The search function definitely searches inside the object themselves, but I have no idea how exactly it works. Regex is also probably not implemented at all, if I had to guess. I’ve also started noticing recently that it fails me sometimes, but I have no idea why. Since I still don’t have that much text inside Anytype, it’s been working fine for me. I would also love if the search function is improved since that is one of the main ways that I search trough my vault.
You can read trough some of these posts to familiarize yourself more with how search works and what can be improved: