The Bible says to give in secret, so does that mean we should not even let people know who it's from, or does it mean we should just not go around saying, "Oh, yeah, I gave this to so and so"? And does that mean for a birthday or a random giving we should not let anyone at all besides us know who it is from? What if we just gave something to somebody when other people were around? Because this seems a bit extreme.
"Take heed that you do not do your charitable deeds before men, to be seen by them. Otherwise you have no reward from your Father in heaven. Therefore, when you do a charitable deed, do not sound a trumpet before you as the hypocrites do in the synagogues and in the streets, that they may have glory from men. Assuredly, I say to you, they have their reward. But when you do a charitable deed, do not let your left hand know what your right hand is doing, that your charitable deed may be in secret; and your Father who sees in secret will Himself reward you openly" (Matthew 6:1-4).
You are getting lost in the figurative language. Jesus states his point: don't be doing nice things for other people to get praise from other people. To emphasize his point, Jesus uses a type of figurative language called hyperbole. That is when you state something in the extreme so that your point becomes clearer. You know figurative language is being used because your two hands don't know what is being done independent of your brain. Since it is an "impossible" statement when taken literally, you know it is figurative.
Many people do good deeds because they want the attention. They don't literally blow horns, but many people will tell others about what nice thing they did to as many people as they can. Such people aren't content with doing good; they want the praise of men.
So Jesus challenges us. Do nice things for other people and don't tell anyone else what you did. Test your personal motives for doing the deed. Now the person you did something nice for might tell other people. That you can't control, but the question is whether you have to "toot your own horn," which is where this cliche came from.
And when someone does something nice for you, you ought to be appreciative and thank them. There is nothing wrong with telling other people what a nice person he was for his kindness. "Let another man praise you, and not your own mouth; a stranger, and not your own lips" (Proverbs 27:2).