I have two different Merriam Webster thesaurus dictionaries, I looked for the word sell as a noun and I couldn't find it, then I looked for the word sale but I did not find the word sell as a synonym, since in the online Merriam Webster dictionary, one of its meanings is "the act of selling, I cannot understand why for native speakers, there is a meaning difference.
As nouns, sale and sell aren't really synonymous. True synonyms are extremely rare in English (if indeed they exist at all! :), so it's not really a useful word or concept anyway.
There are just a handful of "frozen" idiomatic usages involving sell as a noun...
a method of selling in which the person selling tries very hard to persuade the customer to buy something.
a situation where the entire supply of something has been sold
an act of betrayal
sell-off (Stock Market)
the rapid / high-volume selling of bonds, shares, or commodities, usually causing a fall in price
Note that they all differ from the standard noun sale in that they specifically emphasize the act of selling, rather than the broader range covered by sale (most of which focus on [financial] transaction rather than [seller's] action).
“Selling” is an activity that may encompass many discrete acts; it is a noun made from a verb, “sell.”
His new job is selling cars.
It does not refer to any specific transactions.
“Sale” is a noun with several meanings. But the most common is “an act of selling.” When that meaning is intended, it usually refers to one or more transactions that are specific in some context.
The Smiths finally closed on the sale of their house.
Sales of refrigerators are correlated with construction of new housing.
In general, verbs in English do not act as nouns although there are plenty of exceptions (e.g. “contract” although that word is pronounced differently in American English when used as a verb or a noun). As the comments point out, there are certain idioms where “sell” is used as a noun, but usually “sell” is used only as a verb.
Verbs in English can be turned into nouns by adding the suffix “ing” to the root verb. So, “selling” can be used as a noun. When there is a noun with a meaning very close to a verb + “ing,” the two forms are often used in slightly different ways. As I explained above, “selling” can be used as a noun, and “sale” is a noun, but they usually have slightly different connotations.
EDIT: I should add that my answer is limited to American English. I do not know British or Indian English, and my answer may not apply to one or the other or even both.