How does this system handle conflicts?

How exactly, I don't know the details. But there is a built-in "synchronize" feature.

What would happen if the synchorized safe is open on two computers at the same time, they both add a new entry roughly at the same time?

I like the idea of this sync but I don't know how the internal implementation would handle this. Dropbox and Nextcloud both sync the files themselves, and in case of a conflict, they preserve both versions, and let you pick which one you'd like to use as the "canonical". It's a pain but at least the data is preserved.