  We develop the theory of Fra\"iss\'e limits for classes of finite-dimensional
multi-seminormed spaces, which are defined to be vector spaces equipped with a
finite sequence of seminorms. We define a notion of a Fra\"iss\'e Fr\'echet
space and we use the Fra\"iss\'e correspondence in this setting to obtain many
examples of such spaces. This allows us to give a Fra\"iss\'e-theoretic
construction of $(\mathbb{G}^\omega, (\|\cdot\|_n))$, the separable Fr\'echet
space of almost universal disposition for the class of all finite-dimensional
Fr\'echet spaces with an infinite sequence of seminorms. We then identify and
prove an approximate Ramsey property for various classes of finite-dimensional
multi-seminormed spaces using known approximate Ramsey properties of normed
spaces. A version of the Kechris-Pestov-Todor\v{c}evi\'c correspondence for
approximately ultrahomogeneous Fr\'echet spaces is also established and is used
to obtain new examples of extremely amenable groups. In particular, we show
that the group of surjective linear seminorm-preserving isometries of
$(\mathbb{G}^\omega, (\|\cdot\|_n))$ is extremely amenable.
