Hmm, isn't it way easier to say that the sword acts has an identification beacon that allows the mecha to teleport back to its vicinity?

That's just an opinion, but this current explanation relies on teleporting tropes that work better for fantasy settings than sci-fi. This would solve the whole ownership problem and even allow the possibility for beacon arrows to be possible teleport anchors down the line.

Plus it would also explain why the mecha can't teleport to the sword if it's into a backpack or other enclosed space since [url=https://en.wikipedia.org/wiki/Faraday_cage]fully (or even partially, in some cases) encased metal spaces block electronic signals[/url].