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So in the new book we get a picture of the Azyr realm which is a semi-circular ring in space around the remains of the old world.
There's the star, Sigendil (most obviously ripped off of Elendil and Eärendil in LOTR) which shines above.

I'm thinking that the realms are actually different planets rather than dimensions or something of the sort.

There's a mention in the new book of one of the realms looking up and seeing Sigendil, which implies that they exist in the same physical space, and then there's another reference to the gates being used for travel because of the vast distances between them (which you can cross instantly).

Also there's a quote from the author of the first AoS story saying that they were told that each of the new realms was bigger than the old world.

You mean GW 40Ked the fluff of WHFB into AoS. Yeah it has already been pointed out. Endless War.

I believe that somewhere in the fluff section it was written that, stranded on his metal ball, Sigmar found not one but eight worlds, which would make sense as he was drifting through space.

Yes, I'm fairly sure those are planets. Magical ones as you can tell from Sigmar's almost-space-station, but still planets. And the realmgates are literally stargates.