For the purposes of this trait, is the barbarian rage a compulsion effect?
The Rage class feature is not a compulsion effect.
Compulsion effects are normally caused by compulsion spells, which are a subschool of enchantment spells.
Compulsion: a compulsion spell forces the subject to act in some manner or changes the way its mind works. Some compulsion spells determine the subject’s actions or the effects on the subject...
The Rage spell counts as a compulsion effect, because its description classifies it as an "enchantment (compulsion)" spell.
Some class features are compulsion effects, but only if they specifically say so, such as a bard's Fascinate ability. The barbarian's rage class feature does not specify this, so therefore it is not a compulsion effect.
Creatures affected by the Rage spell do count as being affected by a barbarian's rage. However, a barbarian using their rage class feature is not necessarily under the Rage spell.