$\newcommand{\Reals}{\mathbf{R}}$This answer has two parts: 1. Remarks about bijections; and 2. Transfer of operations.
Part I. For definiteness, let's work with finite-dimensional vector spaces with real scalars. Any two vector spaces $(V, +, \cdot)$ of positive dimension contain the same number of elements, and any two vector spaces of dimension at least $2$ contain the same number of vector subspaces.
There are numerous ways to pick families $S$ of subspaces in a way that $|S| = |\Reals|$. Many of these amount to picking "nearly structureless" collections of subspaces. On the other hand, some ways are not entirely uninteresting. Here is a small selection:
In the Cartesian plane $(\Reals^{2}, +, \cdot)$, for each real $t$ let $V_{t}$ denote the line spanned by $(1, t)$, namely the line of slope $t$ through the origin. (The union of these lines is the complement of the vertical axis.)
More generally, by viewing $\Reals^{n+1} = \Reals \times \Reals^{n}$, we obtain an "$\Reals^{n}$'s-worth" of lines spanned by $(1, v)$ as $v$ runs over $\Reals^{n}$. A similar idea gives, in $\Reals^{n+1}$, families of $k$-dimensional subspaces parametrized by $\Reals^{n-k+1}$.
In the plane $(\Reals^{2}, +, \cdot)$, for each real $t$ with $-1 < t < 1$, let $V_{t}$ denote the line spanned by $(t, 1)$. (The union is the set of $(x, y)$ with $|x| < |y|$ together with the origin.)
One natural generalization is to write $\Reals^{n+1} = \Reals^{n} \times \Reals$, where we obtain an open ball's worth of lines spanned by $(v, 1)$ with $|v| < 1$.
In space $(\Reals^{3}, +, \cdot)$, let $S$ be the set of lines on the right circular cone $x^{2} + y^{2} - z^{2} = 0$. Or, let $S$ be the set of planes tangent to this cone. Or, the union of these sets of lines and planes....
Each of these sets is defined by finitely many algebraic equations and inequalities, and a bijection with the set of reals can be given more-or-less explicitly (in fact, is given explicitly in the first two examples, where for the second we use the inverse hyperbolic function $\tanh^{-1}:(-1, 1) \to \Reals$).
More typically, $S$ is an arbitrary collection of subspaces of varying dimension. For instance, let $M$ be a dense subset of the reals such that $M \cap (a, b)$ is Lebesgue non-measurable for every interval $(a, b)$, and let $S$ be the set lines in example 1. above whose slopes lie in $M$. Or, perform some analogous construction in higher dimension. Still, no matter how pathologically we choose, each subspace in $S$ is spanned by some finite collection from our original vector space, so the cardinality of $S$ is no greater than $|\Reals|$, even if a bijection with some $\Reals^{n}$ is inconvenient to write down.
Part II. Suppose $(V, +, \cdot)$ is a (finite-dimensional, real) vector space, $S$ is an arbitrary set of cardinality $|\Reals|$, and $\phi:S \to V$ is an arbitrary bijection. We may define the induced vector space structure $(S, \oplus, \odot)$ by defining
\begin{align*}
U \oplus V &= \phi^{-1}(\phi(U) + \phi(V)), \\
c \odot V &= \phi^{-1}(c \cdot \phi(V)),\quad \text{$c$ real.}
\end{align*}
These equations tell us how to add elements of $S$, and how to multiple elements of $S$ by scalars. Conceptually, the bijection attaches an "avatar" $\phi(U) = U'$ to each element of $S$. We know how to perform vector space operations on avatars, and each algebraic result is itself the avatar of a unique element of $S$. Applying $\phi$ to each side and using primes to denote avatars, the preceding equations may be written
$$
(U \oplus V)' = U' + V',\qquad
(c \odot V)' = c \cdot V'.
$$
It should be clear that $(V, +, \cdot)$ and $(S, \oplus, \odot)$ are abstractly identical (here, naturally isomorphic as vector spaces).
The same idea holds any time we have a set with operations or distinguished collections of subsets, and a bijection with some other arbitrary set $S$.
In example 1. above, the transfer of operations under the bijection $\phi(V_{t} = t$ is particularly simple: The real number indices acts like vectors:
$$
V_{s} \oplus V_{t} = V_{s+t},\qquad
c \odot V_{t} = V_{ct}.
$$
In example 2., the bijection $\tanh^{-1}:(-1, 1) \to \Reals$ allows the induced operation on subspaces to expressed almost as nicely:
$$
V_{s} \oplus V_{t} = V_{\tanh(\tanh^{-1}s + \tanh^{-1}t)},\qquad
c \odot V_{t} = V_{\tanh(c\tanh^{-1}t)}.
$$
