The disjoint union of two $2$-spheres is a compact $2$-manifold. It definitely knots in $\Bbb R^5$ as detected by the linking number. That is, the degree $\alpha$ of $S^2\times S^2\to S^4$, $(p,q)\mapsto \frac{f(p)-g(q)}{||f(p)-g(q)||}$, calling our link $p\sqcup q:S^2\sqcup S^2\to\Bbb R^5$. I'm afraid the linking number is the only invariant here.
A nontrivial link is the Hopf link, whose components are the factors of the join $S^5=S^2*S^2$.

Since $S^2$ unknots in $\Bbb R^5$ (see below), the exterior of one component is homotopy equivalent to $S^2$, and the linking number is also the degree $\lambda$ of $p(S^2)\to S^5\setminus q(S^2)\simeq S^2$. [In different dimensions, where $\alpha$ and $\lambda$ are not numbers but homotopy classes of spheroids (more precisely $\alpha$ factors though a spheroid up to homotopy, upon killing the wedge), their relation is more interesting: $\alpha$ equals the suspension of $\lambda$ (up to a sign).]

Now if $M$ is a connected compact $2$-manifold, then it PL unknots in $\Bbb R^5$; more generally, $k$-connected $n$-manifolds PL embed in $\Bbb R^{2n-k}$ and PL unknot in $\Bbb R^{2n-k+1}$, as long as the codimension is at least three, by Penrose-Whitehead-Zeeman and Irwin. Also by Haefliger (1963), smooth, PL and topological knot theories coincide for smooth $n$-manifolds in smooth $m$-manifolds, where $m>\frac{3(n+1)}2$ (this includes $2$-manifolds in $\Bbb R^5$).

EDIT: More interesting are multi-component links of a bunch of $2$-manifolds in $\Bbb R^5$. Firstly, the isotopy classification reduces to homotopy (=link homotopy) classification. Given a link homotopy, we may view it as a link map of the cylinders $M^2\times I$ into $\Bbb R^5\times I$, and then generically it has at most a finite number of double points. Again all double points can be eliminated by the Penrose-Whitehead-Zeeman trick.

Next, I would think that the homotopy classification reduces to that of $2$-component sublinks. Certainly this is so for $3$-component spherical links, as shown by Massey (Topology Appl. 34 (1990),  269–300). [Note that there are the Borromean rings of three $3$-spheres in $\Bbb R^6$, whose nontriviality is detected e.g. by a nonvanishing triple Massey product in the complement. Thinking of the usual Borromean rings in $\Bbb R^3$ as lying in the three coordinate planes, one can similarly do three copies of $S^1*S^1$ lying in the two-factor subproducts of $\Bbb R^2\times\Bbb R^2\times\Bbb R^2$.]

For multi-component spherical links, Koschorke's $\kappa$-invariant is trivial, as long as all $2$-component sublinks are trivial (Topology 36 (1997), 301-324) and I'm quite sure that in this dimension this should imply that the link is homotopically trivial (hence trivial).