It's very common to remember things wrong, and confuse one memory with another. As far as I can see, there is no official version of Lady and the Tramp with "That's Amore" (though I just did a quick-ish Google search). I think the following things are possible.
1. You first heard/noticed That's Amore and watched Lady and the Tramp at around the same time time, so you remember them together. This is especially possible if you were young at the time.
2. You accociated That's Amore with Lady and the Tramp back then, for whatever reason, and forgot that it was your own thought as you grew up, and assumed it was the real song.
3. You saw a fanmade video with That's Amore as the song.
4. You saw an official video with That's Amore as the song, such as a trailer or one of those music videos that Disney brought out back in the day.