Should a child theme share the same theme options row as the parent, or should it have it’s own options row?

Neither.

Theme options should be stored as theme mods, using get_theme_mod and set_theme_mod.

Internally these map on to options, but it is the official way to store theme specific options, that way when you change themes you don’t get clashes, and your settings are preserved for if you ever change back. It’s the classic adage of “someone else solved that problem for me, it’s their problem now”

But should you persist in using options, I would advise you prefix all your options with your theme name, and not to share options between child and parent. Setting a child themes options should not affect a site when moving back to a parent theme.

There may be cases when it is advantageous though, so exercise good judgement ( aka use set/get_theme_mod )

You can find more about this API here