The operative phrase is "When using rounds of her shifting rage". It works like normal rage, i.e. it's in rounds, you can take Extra Rage to get more rounds, etc.
is the poison of animals based off CL? Or was that save DC?
If they wanted each round of shifting rage to equal a new casting of beast shape there are way too many easier ways of saying so.... But I admit strict RAW are ambiguous here.
The intent, at least, is clear from the introductory text. "...while raging.." in the context of a barbarian is measured in rounds.
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CL is critical for a number of reasons, concentration checks and Dispel checks come to mind. Since this is A Supernatual ability neither one of these might come up. However, this might be an intent to future proof the ability in case some situation comes up (or one that I didn't think of) that involves caster level. Since duration is already specified (rage rounds) the CL is not going to interact at all.
The reference to the Caster Level being equal to her Class Level is simply to cover their asses, it doesn't actually do anything as far as I know.
Shifting Rage alters (rather than replaces) Rage, so unless otherwise noted by the ability it still functions just like Rage. If the intent had been for the ability to have a duration other than 1 round per round of rage spent, Shifting Rage would have had verbiage similar to the Skald's Song of Marching; which states:
"By expending 1 round of raging song, the skald invigorates allies within 60 feet, who may hustle for the next hour... only 1 round of raging song is expended for that hour."
Personally, I think Shifting Rage should have a duration measure in Minutes instead of rounds. The idea of a class built around being a pseudo-lycanthrope who can only change shape for between 30 seconds and 4 minutes and 18 seconds per day (assuming little to no investment in Rage Duration) is just plain stupid. Especially considering that Druids get a very similar ability at 4th level whose duration is measured in Hours instead of Rounds.
Even if a Mooncursed Barbarian got 1/minute per level per "Round of Shifting Rage" spent, they still wouldn't be able to maintain their form as long as a druid before resting.
Can a mooncursed barbarian take Raging Vitality or other feats that have Rage class feature as a prereq?
Yes, because Shifting Rage only alters Rage, without replacing it. That is, the Mooncursed still have the Rage class feature, it just works differently.
In addition to taking Extra Rage to gain more rounds you'll want the Fueled by Vengeance rage power. Every time you damage someone who's damaged you in the last round, you regain a round of rage.
Perhaps the CL being relevant would be for an enemy attempting to use Dispel Magic on the rager?
Supernatural Abilities cannot be dispelled.