The Hadith states that a person that spends freely on his family on the day of Aashura, Allah Ta’ala will increase him throughout the year. This is the maximum that is found in regards to spending on the day of Aashura.
There is no basis for sharing and exchanging gifts in the month of Muharram and particularly on the day of Aashura in Shariah. These are all innovations which ignorant people have included into Deen from their side, thus there is no basis for such actions. Yes, if a person sends a gifts to someone during the month of Muharram or on the day of Aashura as he would have done at any other time of the year without attaching any significance to it, then it would have been acceptable and he would be practicing on the teachings of Rasulullah Sallallahu Alaihi Wa Sallam.
However, now that sharing gifts (on the day of Aashura) has become the trend and custom in our circles and in fact considered a Sunnah, one should refrain from sending a gift on the 10th of Muharram but rather postpone it to a later stage (after the 10th) or gift it at an earlier stage (prior to the 10th), not as a Muharram gift per se, but as a normal gesture of gifting.