@longtresses You’ve tagged your question as language, etc., not philosophy, so I assume you’re asking about English usage, not looking for a debate about metaphysics.
“Recognize”, like “acknowledge”, presupposes that something is already there. You can’t recognize or acknowledge something into existence. So no, with those words it does not go both ways.
The “frenemy” example is a case where the new coinage affects other people’s perception of a phenomenon, but the phenomenon was there before it was named.
Another example: “metrosexual”. Somebody observed some similarities among a set of well-dressed but vaguely effeminate middle-class urban men and came up with a new category to place them in. Perhaps you could say that the category was created when somebody came up with that word, but the men and their similarities were already there (or perhaps it is simply the old category of “fop” with a new word for it). Now, there may be an interplay between the language/thought and the social reality. Men might change their behavior depending on whether they like being classified as metrosexuals. So in that sense it may go back and forth.
Speech-act theory also talks about “performatives”—utterances that create a social reality. E.g., the Queen says, “I dub thee Sir Elton John”, and presto, there is now one more English knight in the world. Or the priest says, “I now pronounce you man and wife”, and the couple is married.
Is there a word in another language that expresses what are trying to say? I still don’t know if any of these examples get at what you mean.