My brother John is clever.

The form in the second example would necessarily be used if you were addressing someone named "John".

Thanks, hightor!

I have been told that with the commas enclosing "John" the second sentence implies that I have only one brother. Is it so?

No. Enclosing "John" in commas, implies you have another brother. For example, My brother, John, is clever, but my brother, Roger, is not.

"My brother John is clever" is the sentence that implies you only have one brother.

Yes, that's how I would interpret it.

The way I understand it is that you can have multiple brothers and still use the same sentence without the commas.

Examples of someone having multiple brothers:

1. My brother John is clever.
2. My brother Steve is clever.
3. My brother Jack is clever.
4. And lastly, my brother Michael is clever.

The use of the commas in the OP examples seems odd and unusual to me.
I might be wrong.
If I am wrong, please correct me.
I've been out of school for a very long time.
My understanding might very well be wrong.

I think I'll have to accept Mame's authoritative ruling here, but if the distinction between having one brother and having several brothers is important you can avoid ambiguity by saying, "My only brother, John, is clever."

I'd like to edit my response, after further thinking. Saying: "My brother John is clever" doesn't exclusively imply you only have the one brother, but that perhaps you are talking about clever people. You could have several brothers, but John is the cleverest, so you would use it in that sense.

Commas, commas, commas!

My brother John is clever.
i.e you have a brother called John who is clever.
My brother, John, is clever.
i.e. You are telling someone called John that you have brother who is clever.