How are the finite flat group schemes $\mathcal{A}[\ell^n]$ arising from an Abelian scheme $\mathcal{A}/S$ singled out among other finite flat commutative group schemes of exponent $\ell^n$?

Is the abelian scheme you consider a fixed one? If the base is of characteristic $p$, and $l = p$ then the Lie algebra of $A[l]$ is isomorphic to $Lie(A)$, so you get a dimension condition. For example, $\alpha_p \oplus \alpha_p$ cannot be embedded in an elliptic curve. There are also more general condition your group scheme $G$ should satisfy. Assume that $S$ is the spectrum of an Artinian algebra and that $G$ is $p$torsion with a trivial binilpotent part. We have an exact sequence: $$ 0 \to G^{mult} \to G \to G^{et} \to 0$$ then the orders of $G^{mult}$ and $G^{et}$ must be equal if $G$ is the $p$torsion of an abelian scheme. On the other hand, in section (15.4) of the book "Commutative group schemes" of F. Oort, there is the following result: Every finite flat commutative group scheme is a subgroup scheme of some abelian variety $A$. 


I'm reminded of two papers by Maja Volkov, an erstwhile student of JeanMarc Fontaine. They are MR2148801 (2006a:14027) Volkov, Maja A class of $p$adic Galois representations arising from abelian varieties over $\Bbb Q_p$. Math. Ann. 331 (2005), no. 4, 889–923. MR1837096 (2002d:11067) Volkov, Maja Les représentations $l$adiques associées aux courbes elliptiques sur ${\Bbb Q}_p$. J. Reine Angew. Math. 535 (2001), 65–101. 

