Arkay is the god of cycle of life and death.
Azura is the Goddess of Dusk and Dawn


Are they the same god?

Azura is the Rim of All Holes - the chinks in the lunar lattice. Arkay is the caretaker of the dreamsleeve, lunar currency getting recycled in an orderly fashion. Azura probably drives him nuts.

Arkay-Aedra.
Azura-Daedra.

Same as Kyne is of nature, as Hircine is to the hunt.

So Aedra can never be same as Daedra?

Several cultures worship Aedra and Daedra in the same pantheon. See the Bosmer's Herma Mora and half the Khajiiti pantheon.

Depends on what you think of Meridia and Malacath...

But does it make them one in the same? The question is not worship, the question is if these entities are the same .

Not in this kalpa at least, thouh On Boethiahs Summoning Day did sort of suggest a celestial game of musical chairs.
The definition is that Aedra are those that particpated in Creation and Daedra did not.
Possibly who is who can vary from kalpa to kalpa, but in the same one an Aedra is not a Daedra.

There are pantheons that venerate both Aedra and Daedra, but not one that suggests an Aedra is a Daedra.

Aedra and Daedra are Aldmeri terms, as seen http://www.imperial-library.info/content/aedra-and-daedra. The main difference between them is that they gave up parts of themselves to create the mortal realm, while the Daedra refused to take part in it. Can they be the same? Not really, but there are a few odd ball cases out there, as seen with Trinimac and Malacath. Arkay and Azura though are completely separate beings. Hell, her artifacts main focus is the trapping of souls, something that drives Arkay bonkers.

No, actually, the question is whether Arkay is Azura. The presumption on your part is that Arkay is definitively Aedric.

I personally think it's unlikely that Azura is the same god as the old man of death and life that the Imperials worship, but be careful with presumptions.