No, they do not mean the same thing. Not only does around make the second sentence grammatical, but the use of for in the first sentence gives that sentence a different meaning.
He has failed the exam for the first time.
He may have written the exam five times before. But this time, the sixth time, he has failed it. It is the first time that he has failed it.
He has failed the exam the first time around.
This means that it's the first time he's written the exam—and that he's failed it.
The sentence construction is a bit odd, but it works if you think of somebody announcing the result immediately after the attempt—and just before he attempts it a second time.
Note the difference that around makes in variations of the sentence:
✔ He has failed the exam the first time around.
✘ He has failed the exam the first time.
✔ He failed the exam the first time around.
✔ He failed the exam the first time.
In the has failed version, the sentence becomes ungrammatical if you simply remove around. However, I can't really explain why this is the case.
The final version without around seems correct, but it is also awkward; I would add he wrote it to the end.