Let $y_i=x_i/c_i$ and $g(y)=f(c_1y_1,\dots,c_ny_n)$. Then the system can be written as $ \frac{dy_i}{dt}=g(y),\quad1\le i\le n.\tag{1} $ If $\{y_1(t),\dots f_n(t)\}$ is a solution, then there exist constants $k_2,\dots,k_n$ such that $y_i=y_1+k_i$, $2\le n$. Conversely, if the real valued function $z(t)$ satisfies de differential equation $ \frac{dz}{dt}=g(z,z+k_2,\dots,z+k_n)\tag{2} $ for some constants $k_2,\dots,k_n$, then $y=\{z,z+k_2,\dots,z+k_n\}$ is a solution of (1).
For any choice of constants $k_2,\dots,k_n$ and $\zeta\in\mathbb{R}$, since $g$ is smooth, equation (2) has a unique solution such that $z(0)=\zeta$. This gives you an infinite number of different solutions of (1).