I have Latin versions of the first two Harry Potter books.
One of Professor McGonagall's instructions, "As quickly as you can, please!", is translated as "quam celerrime potestis, amabo vos!"
Where Harry says "Ginny—don't be dead—please don't be dead—" the Latin text says "Ginnia! ne mortua sis! ne mortua sis, amabo te!"
Has "please" been translated as "I love you"? Were the ancient Romans really that effusive when making polite requests?
"Amo" is used in colloquial Latin without the literal force of "love," in a weakened sense approximating "thank." "Amabo te" is a stock phrase documented already in the comedies of Terence, who wrote about a century before Cicero. Terence's comedies include a number of colloquial expressions, such as "amabo te," meaning "I thank you," almost as an interjection, separated from the syntax of the rest of the sentence.
You can see how such a colloquial use of "amo" might develop over time. In English, we have the colloquial phrase "Love ya!" as a kind of farewell. We don't literally mean "I love you."
"Please" itself in English is a colloquial form of the formal "if it please you" or "if you please (to do something)." Colloquially, the verb alone became separated from the full phrase, and is now used as little more than an interjection, separated from the syntax of the rest of the sentence.
Thus, although it might not seem so on the surface, the history of "amabo te" in Latin and "please" in English are actually quite similar in their linguistic development.