Aonuma made similar (but firmer) remarks back in 2011 about Ocarina 3D and Skyward Sword. Obviously there are numerous story connections between Skyward Sword and Ocarina of Time, such as the link between Ganondorf and the Master Sword being born, the entire 'chosen three' dynamic being set in stone, but there's also smaller stuff like the Harp of the Goddess being the harp Sheik uses in Ocarina of Time.
I'm not really sure what kind of amiibo connection there is, but I'd hope it won't be anything particularly significant. I think it's more likely here we're getting a game that relates in some way to Twilight Princess, though not in ways that mean you miss out if you haven't played TP. As with Aonuma's remarks around Ocarina 3D and Skyward Sword, the suggestion is playing the remake of the earlier game will make you appreciate certain plot elements or points in the new game, and replaying the older game after the new game will have the same effect.
We might actually see the backstory to Twilight Princess played out here, in an early version of Hyrule (so post Skyward Sword), where the 'Interlopers' (who later became the Twili) attempt to use their magic to invade the Sacred Realm?? In that sense, we'd end up seeing familiar elements from Twilight Princess, such as the Fused Shadow and the Light Spirits, without seeing characters like Midna, Zant, the Twili etc who wouldn't exist that far back.