Let $\mathcal{H}$ be a complex seperable Hilbert space. By Kuiper's theorem $U(\mathcal{H})$ is contractible. In particular, it has a different homotopy type from $U(\infty) := \operatorname{colim} U(n)$. The complex Grassmannian of $m$-dimensional subspaces in $\mathbb{C}^n$ can be defined as a quotient \begin{equation} Gr_m(\mathbb{C}^n) = \frac{U(n+m)}{U(n) \times U(m)}. \end{equation} Then the Grassmannian of $m$-dimensional subspaces in $\mathbb{C}^\infty$ is $Gr_m(\mathbb{C}^\infty) = \operatorname{colim}_n Gr_m(\mathbb{C}^n)$.
Question: Let $V \subseteq \mathcal{H}$ be a $m$-dimensional linear subspace. Does the space \begin{equation} \frac{U(\mathcal{H})}{U(V) \times U(V^\perp)} \end{equation} have the same homotopy type as $Gr_m(\mathbb{C}^\infty)$?
As a classifying space, the Grassmannian $\mathrm{Gr}_m(\mathbb{C}^\infty)$ can be modelled as the quotient of a contractible space with free $U(m)$-action by the group $U(m)$. Therefore, in order to show that $$\frac{U(\mathcal{H})}{U(V) \times U(V^\perp)}$$ is a Grassmannian, it suffices to show that the Stiefel manifold $\mathrm{St}_m(\mathcal{H}) := U(\mathcal{H}) / U(V^\perp)$ is contractible with a free $U(m)$-action. Freed's notes proves the contractibility of $\mathrm{St}_m(\mathcal{H})$ by appealing to Kuiper's theorem or the contractibility of the unit sphere in $\mathcal{H}$, and it is not difficult to see that $U(m)$ acts freely on $\mathrm{St}_m(\mathcal{H})$. Therefore, the quotient is a classifying space for $U(m)$.
In the comments, you mention the difference between fixing $V$ or letting $V$ vary. But Freed's notes is not really about varying $V$: he is just being agnostic about which subspace to take. When it comes down to forming the quotient homogeneous space $\mathrm{St}_m(\mathcal{H})$, you take the quotient by the isotropy group of some fixed subspace; it just doesn't really matter which subspace $V$ you choose since the isotropy groups are conjugate.
What may be perhaps a little unsatisfying with this argument is that we use that classifying spaces are "unique". But I think this only extends to showing that they have the same weak homotopy type, so you need a little extra (e.g., CW structures) to show that they have the same homotopy type. In particular, this argument does not immediately provide maps between $U(\mathcal{H}) / (U(V) \times U(V^\perp))$ and the colimit description of the infinite Grassmannian. I believe that this can be done directly, if less slickly.