All of your examples assume that the phrase “the same thing” is taken to be figuratively. That there is some element of “but not exactly the same” attached to the each example.

Show me a single thing, just one single thing that can be repeated absolutely identical, without the "but not exactly the same" attached to it in the real world.

Just one.

Throw a ball twice, so that it hits the same spot with sub-nanometer accuracy. Drop a ball twice in exactly the identical way.

If that's not possible, then it's not possible to do the same exact thing twice.

If that's your premise for the quote, then it becomes "Doing something impossible is the definition of insanity".

The whole quote just disappears into thin air, since it loses all it's meaning.

Hence the "but not exactly the same" is necessary for the quote to even exist.

That's why "doing the same way" never means to do something absolutely identical, because that definition is impossible to ever fulfil. And that's why that definition of "the same" isn't slang but literally in the Cambridge Dictionary

Here are the examples given there:

My twin sister and I have the same nose.

Both noses are identical, down to the atom. Right?

She was wearing exactly the same dress as I was.

Not a single thread is different on the dresses. Down to the atom identical. Cannot be distinguished. Luckily, both of the wearers seem to be of the exact same size as well, right?

Hilary's the same age as me.

Born in the same nanosecond.

She brought up her children in just (= exactly) the same way her mother did.

Not once did she say a different word than her mother said. The education was perfectly memorized and executed absolutely exactly the same way without a single difference. Every word timed to the same millisecond.

Literally exactly the same does not exist outside of mathematics.