Could you please tell me how Hilbert spaces are geometrically linked with our spacetime? Both host functions of $x$, $t$, $m$, ... and there must be a mathematical link between them? A drawing would be welcome.
Second question, related: Is there a space that englobes both?
There is no link. The dimension of the Hilbert space is determined by the number of possible outcomes of experiments measuring commuting observables.
Thus, for a single spin-1/2 system, where the number of outcomes is $2$, the dimension of the Hilbert space is $2$ and it is known one cannot expresse the spin angular momentum operators in terms of spatial coordinates.
On the other hand, for a particle trapped in a 1d harmonic oscillator, the Hilbert space is infinite dimensional.
We only know how to formulate quantum mechanics in flat (Minkowski) spacetime. For this, ignoring spin, Hilbert space uses a basis of position states on a slice of constant time.
