We develop the theory of Fraïssé 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 Fraïssé Fréchet space and we use the Fraïssé correspondence in this setting to obtain many examples of such spaces. This allows us to give a Fraïssé-theoretic construction of $(\mathbb{G}^ω, (\|\cdot\|_n)_{n<ω})$, the separable Fréchet space of almost universal disposition for the class of all finite-dimensional Fréchet 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-Todorčević correspondence for approximately ultrahomogeneous Fréchet 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 $(\mathbb{G}^ω, (\|\cdot\|_n)_{n<ω})$ is extremely amenable.