  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 $(\mathbb{G}^\omega, (\|\cdot\|_n)_{n<\omega})$, 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 $(\mathbb{G}^\omega, (\|\cdot\|_n)_{n<\omega})$ is extremely
amenable.
