The question, as stated, is about the *set* of multiples of translates, but from the example quoted, $\sin x,$ I suspect that OP really meant the *span*.

** Theorem **Let $f$ be a continuous complex-valued function on $\mathbb{R}.$ Then the following conditions are equivalent:

The translates $\{f(x+b) : b\in\mathbb{R}\}$ span a finite-dimensional vector space;

$f$ satisfies a homogeneous constant coefficient linear differential equation;

$f$ is a finite linear combination of functions $f_{k,\lambda}(x)=x^k e^{\lambda x}.$

*Proof.* If $f$ is assumed infinitely differentiable then all derivatives of $f$ belong to the $\mathbb{R}$-span of translates of $f.$ Thus condition 1 implies that $f$ and its derivatives of order up to $n$ are linearly dependent over $\mathbb{R},$ which is condition 2. The smoothness assumption may be removed by using the Fourier or Laplace transform.

The equivalence of conditions 2 and 3 is a basic fact of ODEs. Finally, a direct computation shows that $f_{k,\lambda}(x)$ spans the $(k+1)$-dimensional vector space $\{P(x)e^{\lambda x}: P\text{ is a polynomial of degree} \leq k\}$, so condition 3 implies condition 1. $\square$

Condition 1 – 3 have the following representation-theoretic interpretation. The additive group of $\mathbb{R}$ acts on itself by the right multiplication. This gives rise to a linear representation of $\mathbb{R}$ on the functions on $\mathbb{R}$ via translations called the *right regular representation*, and condition 1 states that $f$ belongs to a finite-dimensional subrepresentation $V$. Finite-dimensionality of $V$ implies that $V$ contains an irreducible subrepresentation $W$, which must be one-dimensional (Schur's lemma), hence $W$ is spanned by a character of $\mathbb{R}.$ All *continuous* characters are the exponential functions $e^{\lambda x}$ for various $\lambda\in\mathbb{C}$; however, using a Hamel basis of $\mathbb{R},$ it is easy to see that there are uncountably many others.

Condition 2 is the Lie algebra analogue of condition 1: viewing $\mathbb{R}$ as a one-dimensional Lie group, the content of Lie's theorem is that its finite-dimensional (continuous) representations correspond (by differentiation and exponentiation) to f.d. representations of the abelian one-dimensional Lie algebra, i.e. to a single linear transformation on $V.$ The span $V_{n,\lambda}$ of the functions $f_{k,\lambda}$ with $0\leq k\leq n-1$ from condition 3 is an $n$-dimensional indecomposable representation of $\mathbb{R},$ whose infinitesimal version is a Jordan block of order $n$ with $\lambda$ on the diagonal. Moreover, any subrepresentation isomorphic to $V_{n,\lambda},$ i.e. corresponding to the same Jordan block, must be $V_{n,\lambda}$ itself.