The existence of an inverse (and of a neutral element or identity with respect to which it is defined) is the fact that guarantees that we can solve an equation of the form: $a*x=b$ and find $x=a^{-1}b$.
This seems a good reason to deserve a special name ( and attention) to such a structure.
In other words: the existence of a neutral element $e$ and an inverse $a^{-1}$ make a group the simpler structure in which we can solve the equation:
$$
a*x=b \Rightarrow (a^{-1}*a)*x=a^{-1}*b \Rightarrow e*x=a^{-1}*b \Rightarrow x=a^{-1}*b
$$
At a certain level, an $\infty$-group $G$ is not so different from the groups you are used to. $G$ has an underlying space, analogous to the underlying set of a group, a unit $e:*\to G$, a multiplication $m:G\times G\to G$, and an inversion map $i:G\to G$. Where things begin to get interesting is the sense in which these operations satisfy the group axioms. It is no longer true that $m(m(g_1,g_2),g_3)=m(g_1,m(g_2,g_3))$; rather, this associativity axiom becomes yet another piece of structure, an associator map $a:G\times G\times G\times I\to G$ giving a homotopy between the two sides of the above equation. Similarly, we have to introduce "unitors" replacing the equations $m(g,e)=g$ and $m(e,g)=g$ with homotopies, and similarly for the equations involving the inverse.
Furthermore, this isn't all! In fact a group satisfies many more equations than those in the usual axiomatization. For instance, there are five different ways to parentheses four letters: $g_1(g_2(g_3g_4)),(g_1g_2)(g_3g_4),((g_1g_2)g_3)g_4),(g_1(g_2g_3))g_4,$ and $g_1((g_2g_3)g_4)$. In a group, these are all equal, and this follows from the associativity axiom. In an $\infty$-group, we have homotopies between these five parenthesizations, interpreted as maps $G^4\to G$. In fact, we can paste these homotopies together into a map $G^4\times \partial P\to G$, where $P$ is a regular pentagon in the plane. We would like to know that there is, in essence, only one way to associate two products into each other-it would be a bad generalization of group theory if we could follow a nontrivial loop in $G$ by simply associating one word back to itself in some complicated sequence! Thus part of the structure of an $\infty$-group is an extension of the above map to the pentagonator $\pi:G^4\times P\to G$.
And we're not done yet. In fact, there are infinitely many levels of structure needed for describing all the ways of associating longer and longer words alone, and the spaces which are $I$ for the associator and $P$ for the pentagonator continue growing in dimension and combinatorial complexity. Stasheff gave the first complete description of this part of the structure of an $\infty$-group, which he called an $A_\infty$-space, for a space with a multiplication which is Associative up to a homotopy which is itself well defined up to a homotopy which is well defined up to...Stasheff's original papers are still excellent reading on this topic.
The nLab's "group object" in an $\infty$-category is closely related to Stasheff's notion of $A_\infty$-space and in similar ways to the elementary notion of a group. The "simplical object in an $\infty$-category" that is the underlying structure of a group object is supposed to represent the group $G$ together with all its finite powers $G^n$ (including $n=0$) while the simplicial face maps correspond to the canonical projections between $G^n$ and $G^m$, the degeneracies correspond to various ways of mapping $G^m$ to $G^n$ by inserting copies of the unit, and the various pullback squares cleverly encode the multiplication, inverse, and all the infinite tower of homotopies witnessing the axioms as in the previous paragraph. This is also closely connected to Lawvere's perspective on groups: a model of the Lawvere theory of groups in a category $C$ is exactly all the stuff I just said, except that the homotopies are allowed again to be equations. So it's no different than an ordinary group object, except insofar as we don't pick out particular operations and axioms as privileged.
This is a pretty complicated structure! A large amount of work in the last fifty years of algebraic topology has been on how best to understand these objects. One fundamental theorem is that an $A_\infty$-space has a delooping if and only if it is actually an $\infty$-group: having a delooping means it is homotopy equivalent to the space of based loops in some pointed connected space, which is itself unique up to homotopy equivalence. And a map of $\infty$-groups, i.e. some kind of homomorphism appropriately preserving all the huge mess of structure up to a huge number of homotopies, is nothing more than a map of their deloopings. (This statement is a little bit cleaner than the reality, but it's close.) This is the equivalence between $\infty$-groups and pointed connected objects you mention. This has less of an obvious analogue to ordinary group theory, but it's still there: it's simply the well-known perverse definition of a group as a groupoid with a single object. The reason the equivalence is so much more interesting in $\infty$-category theory is that pointed connected $\infty$-groupoids, i.e. pointed connected spaces, are usually not defined with algebraic operations of composition and inversion of loops, so much less structure has to be carried around in defining them and in particular their maps. Another simplification is that every $\infty$-group is appropriately equivalent to the geometric realization of a simplicial group, that is, a completely ordinary group object in the ordinary category of simplicial sets. At least the equivalence between pointed connected spaces and simplicial groups is the oldest of all these results-it goes back to Kan in the early '60s.
Anyway, hopefully that's given a bit more of an idea of what's going on. There are many approaches to the concept, largely because all of the approaches become intolerably complicated in one way or another. This situation is characteristic of $\infty$-category theory, and in the current state of knowledge it appears unavoidable that appending "$\infty$" to a familiar object creates substantial complications.
Best Answer
No, these are very different structures with very different requirements.
An operation is different than a relation. An operation is a function $A \times A \to A$ while a relation is simply a subset of $A \times A$. We can think of any relation also as a function $A \times A \to \{0, 1\}$. This way it becomes obvious that an operation is not the same as a relation.
To be more concrete for us to have a group noone requires from us to have ordering (even partial) between the elements of the group. While for a partially ordered set this ordering requirement is essential.