My concern here is not with a dictionary definition, but the meaning of "accompaniment" in the specific context of contemporary papal teaching—what the term meant to John Paul II, what it means to Pope Francis; how we, the faithful—priests and laity alike—are to receive it and live it.
It's not a pat answer, so bear with me.
I first came across the concept reading George Weigel's magisterial biography of John Paul II, Witness to Hope, shortly after its publication in 1999, on p.100, where Weigel is recounting the period of the Fr. Wojtyla's university chaplaincy in Krakow in the 1950s.
Previously, the chaplain's task had been to provide sacramental services to students. Wojtyla, who intensified the chaplaincy's sacramental ministry and involved students in it liturgically, thought of his chaplaincy as a ministry of "accompaniment," a way to "accompany" these students in their lives. The chaplain's presence couldn't be limited to the sanctuary and the confessional. A really effective chaplaincy, he believed, had to be present to these young lives in the world as well as in the church.
I fell in love with the idea immediately and have cherished and brooded over it ever since. That spontaneous response in me has something to do with my particular background, which is part of a larger pattern I see almost everywhere I turn in the Church and in society these days. So, at the risk of being too self-referential, let me explain a little.
The formation of my soul involved a large element of authoritarianism in various guises and modes. There was sternness and staunchness in my upbringing; many rules, demands and expectations, and too little in the way of affection, closeness, and encouragement. I grew up "good", morally and religiously, but shy and insecure, full of anxiety, self-doubt, and self-reproach.
The friendship of evangelicals in my adolescence—the consciously personal and affective character of their spirituality—and then later, the warmth and liveliness of the religious atmosphere of my alma mater, saved me. Most importantly, I learned, slowly and spastically, to cultivate the interior life. I grew to understand that holiness was much more about intimacy with God, about love and self-giving, than obeying the law or living up to an ideal.
But that was not the only thing I learned at my alma mater, alas. Mixed in with all the warmth and joy and genuineness of the religious atmosphere of the Franciscan University in the late 80's was a regrettable and damaging element of authoritarian control, which came through its association with the Covenant Communities of that day. There was a lot of sincerity in the communities' attempts to embody in policies and customs a way of life grounded in the experience of grace they'd received through the charismatic renewal. But, it went off the rails somewhere along the line, and eventually bishops had to intervene and force reforms.
That story is for another time and place. Here I will only zero in on one central element, for the sake of the light it throws on the meaning of "accompaniment" in contemporary Catholic life and thought.
In those communities and on our campus—no doubt in reaction to the laxity and relativism then sweeping the wider culture—the role of authority and obedience was stressed excessively. While there was much preaching and teaching about God's love for us personally, very little was said or taught about the prime importance of conscience and self-determination in the moral life. Conscience was viewed as something suspect and dangerous; we students were deliberately "formed" to "submit to authority." "God always blesses obedience." And by that was meant not only obedience to the objective moral law or the positive obligations of our Faith, but obedience to a spiritual director, your "pastoral leader" (who was typically another layman in the community), the residence hall director, your household coordinator, the student life staff, and to the rules that constantly proliferated, both spoken and unspoken. There were rules regarding how to dress, whether and whom to date, how much money you needed to be earning before it was okay to propose to your girlfriend, how children should address adults, how wives should speak to their husbands, whether men should change diapers, and on and on. We weren't punished for breaking the rules. It wasn't like that. It was a matter of indicating how serious we were about wanting to please God.
We were taught to mistrust ourselves and our own powers of discernment. The path to holiness lay in handing over more and more of our decisions to someone else—someone "holier" and above us in the communal hierarchy. And meanwhile, the "leaders"—those in positions of authority—were taught to believe that it was their role to control others' lives. That was how they served God and the community.
Most of us meant well, but we were under a sort of spell. The encounter with personalist philosophy snapped me out of it, though the bad habits of externalism, judgmentalism, and self-doubt I'd acquired took many years to unlearn.