  We realize the noncommutative Gurarij space $\mathbb{NG}$ defined by Oikhberg
as the Fra\"{\i}ss\'{e} limit of the class of finite-dimensional $1$-exact
operator spaces. As a consequence we deduce that the concommutative Gurarij
space is unique up to completely isometric isomorphism, homogeneous, and
universal among separable $1$-exact operator spaces. We also prove that
$\mathbb{NG}$ is the unique separable nuclear operator space with the property
that the canonical triple morphism from the universal TRO to the triple
envelope is an isomorphism. We deduce from this fact that $\mathbb{NG}$ does
not embed completely isometrically into an exact C*-algebra, and it is not
completely isometrically isomorphic to a C*-algebra or to a TRO. We also
provide a canonical construction of $\mathbb{NG}$, which shows that the group
of surjective complete isometries of $\mathbb{NG}$ is universal among Polish
groups. Analog results are proved in the commutative setting and, more
generally, for $M_{n}$-spaces. In particular, we provide a new characterization
and canonical construction of the Gurarij Banach space.
