Why the alphabet of the digital information is composed of 2 elements?
You are really asking three different questions:


Why are computers based on binary?
Why did Boole use a binary system?
What does all of this have to do with entropy?


The answer to the first question, as you suspect, relies on implementation details; see for example this question. In other circumstances, such as wired or wireless communication, information is transmitted in larger chunks, what is known as modulation.

The answer to the second question is simpler: Boole was formulating the laws of thought, that is logic, in which each statement is either true or false. That's your binary distinction.

Finally, I can't really follow your third question. According to Shannon's source coding theorem, you can optimally compress information using any channel. Text is usually stored as text since it's easier to process that way, but we can compress it to reduce its size to roughly the entropy. In some sense, text is stored using an alphabet of size roughly 64, including lower and uppercase letters, digits and punctuation marks. Other data is stored in bytes, which is an alphabet of size 256. All of this has nothing to do with entropy, rather it's ease of processing. These are just some thoughts, since I can't really understand this question.