I've collected the following Daedric Artifacts on the same character without getting the achievement:

Mehrunes' Razor

Dawnbreaker

Ebony Blade (I tried this fully charged and uncharged)

Ring of Namira

Black Star

Ring of Hircine

Masque of Clavicus Vile

Ebony Mail

Wabbajack

Volendrung

Skull of Corruption

Sanguine Rose

Mace of Molag Bal

Spellbreaker

Oghma Infinium

I've read that if I am able to find Sinding I may still be able to get his hide, however I am unable to find him. I have attempted to equip all items except for Oghma Infinium which I have already read. The latest patch was supposed to fix this issue however, so I don't think that reading it had an effect.

If anyone has any idea, please let me know.

Found a solution: Sinding was somehow back in Falkreath prison. Wore him down with destruction as he was behind prison bars. When he was close to the bars I finished him off so that he was close enough to search his body. Was able to take his hide and gain the achievement. Still, a glitch that Bethesda might consider fixing in any further patches.

This achievement is bugged if you have Dawnguard installed. The DLC overwrites the property that triggers the achievement counter for "A Daedra's Best Friend" after obtaining the Mask of Clavicus Vile. If you are on PC, the Unofficial Dawnguard Patch fixes this. Otherwise, the only "workaround" is to get both the Ring of Hircine and the Savior's Hide. Unfortunately, Bethesda isn't doing any more patches and they never have bothered to patch their DLC.