  We are concerned with the focusing $L^2$-critical nonlinear Schr\"odinger
equations in $\mathbb{R}^d$ for $d=1,2$. The uniqueness is proved for a large
energy class of multi-bubble blow-up solutions, which converge to a sum of $K$
pseudo-conformal blow-up solutions particularly with low rate $(T-t)^{0+}$, as
$t\to T$, $1\leq K<\infty$. Moreover, we also prove the uniqueness in the
energy class of multi-solitons which converge to a sum of $K$ solitary waves
with convergence rate $(1/t)^{2+}$, as $t\to \infty$. The uniqueness class is
further enlarged to contain the multi-solitons with even lower convergence rate
$(1/t)^{\frac 12+}$ in the pseudo-conformal space. The proof is mainly based on
the pseudo-conformal invariance and the monotonicity properties of several
functionals adapted to the multi-bubble case, the latter is crucial towards the
upgradation of the convergence to the fast exponential decay rate.
