In English language, the three words are assigned multiple meanings. They don't share all of their meanings but they share at least one meaning in common. In this sense, they are synonyms.
SAME
The word "same" is often, but not always, used to indicate absolute sameness.
"These two shirts are the same."
This means they are two shirts, not merely one shirt, that are absolutely or completely the same ( i.e. they are the same in every detail, not merely in certain way. )
The word is also sometimes used to indicate that the thing has the same identity as some other thing.
"This is the same shirt I wore yesterday."
This means, "This shirt has the same identity as the one I wore yesterday." In other words, it means, "They are one and the same shirt."
The word is also used to indicate relative sameness, e.g. "They are the same shirt in terms of their color."
IDENTICAL
The word "identical" is almost always used to indicate absolute sameness. It can also be used to indicate relative sameness.
"The two shirts are identical."
This means they are exactly alike in every detail.
They are not merely similar. They are not merely the same in some limited aspect. They are absolutely the same,
And the statement also does not mean that they have the same identity. In other words, it does not mean they are one and the same object.
However, in a small number of cases, and almost exclusively within philosophy, the word "identical" is used to mean "has the same identity as" or "one and the same as". That's why 20th century philosophers decided to introduce the terms "numerically identical" and "qualitatively identical" to distinguish between the two concepts. They defined "numerically identical" to mean "one and the same" and "qualitatively identical" to mean "absolutely the same".
EQUAL
Used to indicate absolute and relative sameness. Most often used within mathematics.
"Sets A = { 1, 2, 3 } and B = { 1, 2, 3 } are equal."
This means the two sets are absolutely the same.
CONCLUSION
Learn English language.
But more importantly, pay attention to what people you are desperately trying to criticize are actually saying so that you don't end up strawman-ing them.
Try to listen sometimes instead of dogmatically forcing your beliefs onto others.
Train yourself to be open-minded and a good interlocutor instead of pretending that you already are.