Let me suggest that the obsessio of both Catholicism and Orthodoxy is Union with God. This is most clearly seen the the contemplative and monastic traditions which provide each tradition a vision of the spiritual life. Relatedly, in both traditions sanctification has a greater emphasis than it does in Protestant spaces. Also, visions of salvation describe mystical union, theosis, divinization, and the Beatific Vision. Salvation is more ontological, a union, than a forensic emphasis on forgiveness.
Both traditions also emphasize mystical encounter in the Eucharist. Where Protestants "go to church" for a sermon, Catholics and the Orthodox go to church to witness a miracle, to come into the "real presence" of Christ and to incorporate, through the meal, Christ's power and life. Eucharistic adoration is an associated practice. Being in physical proximity to God, an experience absent in the Protestant imagination, is crucial to the devotional life.
There is also the tradition and expectation of divine appearances and apparitions, of both Christ and Mary, along with sites associated with miraculous healings.
In short, due its contemplative tradition, vision of salvation, sacramental life, divine apparitions, and sites of healing, to name only a few things, the epiphania of Catholicism and Orthodoxy is a direct encounter with God and deepening union with God. Given this, it's no surprise that mystically inclined Christians, driven by the obsessio of Union with God, tend to gravitate toward these traditions.
While Union with God is the main obsessio of Catholicism and Orthodoxy, let me add two others.
The obsessio of Death is also found in Orthodoxy. As I describe in The Slavery of Death, for the Orthodox our Fall from Paradise is more associated with Death than Guilt, causing Orthodoxy to emphasis Christus Victor visions of salvation over forensic views, such as penal substitutionary atonement. For the Orthodox, the Incarnation and the Resurrection are the critical aspects of salvation, given how they overcome the ontological predicament of death, in contrast to the Protestant emphasis, given its obsessio of Guilt, upon the cross.
I would also suggest that Belonging is the obsessio of many who convert from Protestantism to Catholicism or Orthodoxy. To be sure, these conversions are driven by the obsessio of Union with God, people searching for the "real thing," by way of divine encounter when it comes to the sacramental life, in contrast to the hollow, superficial, thin, and even counterfeit, spirituality they find in Protestant spaces. But there is also a longing to join the Mystical Body of Christ, which fuses Union with God with Belonging. Converting to Catholicism or Orthodoxy is experienced as a "coming home." In contrast to the evangelical obsessio of Belonging, which gets linked with the obsessios of Guilt and Self-Alienation, the Catholic experience of Belonging tends to be more mystically ecclesial than individualistic.
