If you're compactifying a moduli space by choosing degenerations, you typically assign the trivial log structure to the uncompactified space. The size of the characteristic at a boundary point roughly describes how degenerate the corresponding object is, and in a place where the log structure is trivial, the characteristic is the trivial monoid.

When someone says that the log structures on the moduli space of marked curves and the tautological curve over it are unique, that is relative to some condition that needs to be specified, e.g., being an essentially semistable morphism. If that condition is assumed, then the log structure is unique. In the case of marked curves, the locus of schematically smooth curves is then given the trivial log structure.

I don't know what you mean by unique deformation. The tangent and jet spaces of a smooth compactified moduli space are just as big on the boundary as they are elsewhere.

If you're compactifying a moduli space by choosing degenerations, you typically assign the trivial log structure to the uncompactified space. Given a point $x$, the size of the characteristic $M_{X,x}/\alpha^{-1}\mathscr{O}_{X,x}^\times$ roughly describes how degenerate the object over $x$ is, and in a place where the log structure is trivial, the characteristic is the trivial monoid.

When someone says that the log structures on the moduli space of marked curves and the tautological curve over it are unique, that is relative to some condition that needs to be specified, e.g., being an essentially semistable morphism. If that condition is assumed, then the log structure is unique. In the case of marked curves, the locus of schematically smooth curves is then given the trivial log structure.

I don't know what you mean by unique deformation. The tangent and jet spaces of a smooth compactified moduli space are just as big on the boundary as they are elsewhere.