I am wondering what is meant by "Arena"? For example in the lore book Brothers of Darkness about the Dark Brootherhood there is a passage that reads something like "at a time when the Arena was a fractured land". Is Arena the name of the planet the Elder Scrolls games are set?

It was the name of Tamriel when The Elder Scrolls: Arena came out, and it's a bit of lore kefuffle.

As the first game of the series, Arena was idealized as a gladiator sim. When they changed gears, they weren't able to come up with a new title. Therefore, they mention Tamriel, and then say it's called Arena to justify the title.

In short:

Nirn: Planet of the Elder Scrolls takes place on.

Mundus: What we consider "reality."

Tamriel: The continent the Elder Scrolls takes place.

Arena: Holdover from the first game, rarely used to refer to Tamriel outside of Arena, and perhaps Daggerfall.

Well, was more to do with the fact that when Arena (The game) changed it's focus, most of the promotional material, box art and other related stuff had already been done.

So, it was a case of Arena in this sense being used in the same way we'd use 'Combat Zone' or 'War Zone' But otherwise, yeah, as you say pretty much a clumsy way of ensuring the title made sense at the time.