Translation:Even though they like tea, they don't drink tea.
Literally "nose on my mouth." It's an ancient idiom. The earliest instance I know of is Talmud Bavli, tractate Chagiga, folio 3a. The word picture is that of the nose enjoying a higher physical position on the face, and thus is seemingly of higher status, "even though" the mouth more often takes the spotlight.
They don't quite mean "exactly the same thing" (although they obviously share a lot in common: "though"!). "Even" is an additive, for example, that can be used to add or affirm something that would otherwise be unexpected, or sometimes even(!) "least" expected, relatively speaking. It's often used to add something and intensify it (sometimes at the level of words and list-items, or even(!) at higher levels like clauses, sentences, etc.).
For example, "I like broccoli, cucumbers, spinach, and even asparagus." Alternatively, "I like broccoli, cucumbers, and spinach. I even like asparagus."
Here are a couple examples related to the above translation, which shows that the addition of "even" slightly changes how we construe (or conceptualize) these sentences (even though the basic conceptual content is still the same):
"Though I like tea, I don't drink tea." "Even though I like tea, I don't drink tea."
"Though I dislike fish, I eat fish." "Even though I dislike fish, I eat fish."
Stephen Levinsohn, a linguist and translator, details some of the cross-linguistic uses of such connectives, for example, in "Self-Instruction Materials on [Narrative and Non-Narrative] Discourse Analysis": http://www-01.sil.org/~levinsohns/discourse.htm
"Although according to them, they love tea, they are not drinking tea." Is this wrong, or is this machine checking that just doesn't think of correct alternatives? I am not an EXPERT, obviously, but it seems that על פי is frequently used attributively for - at the mouth of - or according to what they say?
First, this is not machine checking. The right answers are input and the computer will mark your answer as correct if it fits one of the input answers or a close typo.
Second, you are over-thinking this. Although על פי means "according to" and אף על פי could be interpreted as "even according to", that is not what the sentence is saying. The three words אף על פי mean when they're together "although" or "even though". The is no "according to them" in that sentence. The first part of the sentence means "even though they like tea", nothing else.
As long as you follow any of them with something beginning with "ש" they are interchangeable:
למרות שהם אוהבים תה, הם לא שותים תה
אף על פי שהם אוהבים תה, הם לא שותים תה
אפילו שהם אוהבים תה, הם לא שותים תה
The last one is less common, because אפילו is more commonly used for other things:
אפילו אני שותה תה
Even I drink tea. Because that is what אפילו means -- even. And by the way, the אף in אף על פי is just short for אפילו, but don't ever say "אפילו על פי" - that is not idiomatic.
The word למרות can be followed by a noun or a noun phrase, which the others can't:
למרות אהבתם לתה, הם לא שותים תה
Despite their love for tea, they don't drink tea.