According to Merriam-Webster:
to gain knowledge or understanding of, or skill in, by study, instruction, or experience
also:
Memorize
To explore your sample sentence:
I learn English every day
could be a contracted version of
I listen to people talk to each other all day long, and I'm often able to identify new words and language constructs, that I will even remember later. In this manner, I learn [some] English every day.
Now it's established that the person has succesfully retained the knowledge. Therein lies the key difference.
Compare it with:
I study English every day.
I sit over the books for several hours, but I often find that I have to repeat lessons because some of it does not stick.
Studying
just gives a testimonial about the effort made. But it does not even try to deal with the part whether the effort was fruitful or not.
Now to your ultimate sample sentence:
I learnt English. It doesn't mean I studied English.
I feel this could be said by someone who is full of themselves, characterized by self-pride. In this scenario this person offers a declaration of their superiority, or at least of exceptional nature.
A different version could be:
A am so talented that I did not need to make significant conscious efforts at studying (sitting over the books for several hours). Thanks to my talents, I picked it all up just by haggling with salespeople at the market.
Just addressing your last question:
Does the sentence with "learn" sound as I learn all English forget it during the night and learn it again the next day?
As you have seen from the above, exactly not. The case is exactly the opposite.
Studying might as well mean an effort that needs repeating.
Learning (implied: successfully) however needs to be achieved only once; as soon as it's achieved, the person is ready to use the retained knowledge for gainful activity :)