The Companions do not have an alternative/different/unique strain of Lycanthropy. They are different only in the method by which they were inflicted with the disease. The other differences that can be observed from previous games; the lack of forced lunar transformations, seemingly new self control, lack of the need to kill, all have explanations that don't involve the Companions being special.

Lunar transformation: Neither The Companions nor the PC undergo lunar transformations because for any conceivable reason from negligence to game direction, the devs left it out. That does not mean that they don't transform behind the scenes. Typical gameplay/lore segregation. As for the ability to transform once a day that the PC has, the same power was in Daggerfall and I would guess that other beast have the ability as well, as seen with Vilkar?, Aela and Sinding.

Self control: Some would argue that the Companions are special because they have control over their beast forms but we are assuming that Werebeast don't have control in beast form. Some Werebeast go mad and become feral and crazy but I believe that there is precedence for control as well. In Daggerfall, the rouge Wereboar clearly had some sense of control if it was defying Hircine. The Werewolves in TES:III Bloodmoon were intelligent and self aware enough to carry out more or less complicated tasks and Hircine himself stated that guile was a trait that he valued in hunters. The Werebeast, both wolves and bears described in the TES novel "The Infernal City" did not seem to be mindless beast. In Skyrim, Sinding has some control and can even speak but he has no apparent association with The Companions. It is my view that The Companions, despite their faults, provide a greater view into the lives of Werebeast.

Need to kill: Again, gameplay/lore segregation. The Companions probably don't go out and kill (full moon) because it could present problems the devs didn't want to deal with.

I think that the most significant argument I have is the fact that no one, The Companions included, evey actually state that the Companions are special or different in any way. No where have I been able to find this information. The only concrete difference is how they came about their Lycanthropy: The Glenmoril Witches. I don't see how receiving the disease from the Coven would make the Lycanthropy different. Hircine has always had a unique relationship with this particular Coven and they have often held the key to the curing of Lycanthropy so it makes sense to me that Hircine would extend his blessings through them. I think the heart of why Werewolves (and the lack of Werebears) are different from earlier games is a combination of game direction and time management. Skyrim took a different direction than the previous TES games, however slightly, as is evident in the removal of attributes and etc, combine that with the theories of little time and implementation given to Werewolves and Vampires that some of us have and thus you have the Werewolves of Skyrim. Not truly different, just unrefined, hopefully only for now.

I've never heard anyone say the companion strain was anything different.

I'm not sure I get the point here.

I agree with this point, especially with, as you said.

I think they just use their beast form enough that they don't unwillingly transform during full moon, have control over your beast form and all that other stuff. The first time you transform, it's implied that you black out after a couple seconds after transforming and lose control.

The Companion strain is (by technicality) not true beast blood. They have the affliction due to magic users (Glenmoril Witches) and not directly by Hircine hence why the Companions have such control and are not afflicted by the full moon.

Plenty of lycans go feral but not all. Those (kinda) like Sinding have control of their inner wolf (or those that possess Hircine's Ring) are affected by the moon.