I would argue that Engels was correct in his interpretation. Funnily enough this question coincides quite closely with part of my master's thesis. In it I argue, based on the theoretical framework of Michel de Certeau, that the early labor movement can definitely be seen as a continuation of religious practices, or perhaps more accurately as an appropriation of them. My original draft for this answer included an introduction to Certeau and his broader ideas about modernity and faith. This became quickly bloated and off topic though, so instead I will answer this by briefly discussing the religious significance of the Bible in the Middle Ages before going into how Das Kapital in a sense strived to fill this same position in the late nineteenth century. I will implicitly be using Certeau as the basis for my answer, so keep in mind this isn't meant as some definitive truth on history as much as it is one way of looking at things. As an aside, I will be using socialist to refer mostly to what we would now call the social democrats of late nineteenth century Europe.

Now firstly, the average medieval peasant would not have read the Bible for themselves. Not only were literacy rates generally quite low, but at this time the Catholic Church still strongly insisted on keeping the Bible text solely in Latin thus further thinning the herd. This meant only those educated in reading Latin, in practice the clergy, could actually read the Bible for themselves. Obviously the Bible still had a large impact on the lives of the average peasant despite them not being able to read it. This meant the Church played a vital role throughout this period in guaranteeing the validity of Scripture on the one hand, and in ensuring its commandments are being followed on the other hand. In essence, the Bible and the writings of the Church Fathers were seen as the reflection of the cosmic harmony. From these words derived the medieval worldview and the legitimization of the existing social hierarchies. The peasant was not to dream, for the social order and his position on it are ordained features of God's order. Just keep your head down and follow what the Church says and you will be rewarded with an eternity of bliss in the afterlife.

Now at some point, people started to become distrustful of the Church, and I'd say rightly so. Their position as the sole interpreter of the Bible was thus being questioned, and eventually culminated in the Protestant Reformation. This idea that believers should no longer just believe what the Church told them but read the Bible for themselves coincided with the rise of another fundamental building block of modernity: the printing press. These two factors collided and reinforced each other as the medieval world view slowly started crumbling and the rise of capitalism was slowly becoming more apparent as a new class of people, the bourgeoisie, started to appear on the stage. The modern man, in contrast to their predecessors from the 'Dark Ages', no longer was destined to undergo an order prescribed by God. Instead, he is meant to create his own meaning in this world. To do this, he no longer must listen to what the Scripture says, but instead perform his own scriptural operation. From the blank page he must construct a new order to believe in.

Now I'm obviously condensing a lot of history into very few words here, but this idea of modern man creating his own meaning in the world is what underpins modernity according to Certeau. It manifests itself in an ongoing obsession with categorizing the world, with trying to explain and thus over time creating entire models of how the world works. These models are then to be used to alter the given world and shape it to the whims of their writers. It is also the basis of bureaucracy, which Certeau for example examines by researching the French Revolution and how its proponents used the French language as a unifier, but also as a tool to control and to manage resistance. But perhaps most importantly for this answer, it is also intrinsically linked with the rise of capitalism. The scriptural economy, as Certeau calls it, is ever-expanding, seeking to incorporate ever more of the world into its project of not only understanding, but more importantly using it for its own gain. Just as the Bible before it, our writings (whether scientific, historic or even more mundane, that of mass media) seek to constitute an order by which the world should be organised, by understanding it we are meant to be better equipped to engineer it to the goals of those in charge.

In The Practice of Everyday Life pt. 1, Certeau describes how this scriptural operation also came to underpin the socialist movement. In a passage where he's certainly not scared to draw the parallel with Christianity, he explains how the socialist movement also became entangled in the scriptural economy, and how over time their institutions more closely started to resemble that of the Church. At the core for Certeau here is the aspect of faith, and the idea that the socialist movement attempted to cultivate faith within the proletariat. For Certeau, the future is an essential part of faith: a believer makes sacrifices in the present believing that they will be rewarded in the future. The Marxist worldview is obviously also strongly oriented toward the future, yet in a subtly different way from its religious predecessor by no longer focusing on a metaphysical afterlife but instead material reality. The worker must sacrifice now, fight and risk reprisal, because in the future a complete revolution is inevitable. This future-oriented approach can even be seen in the writings of what we consider the predecessors of figures like Marx, aptly named the utopian socialists.