I would say, monomials are not a Shauder basis for $C[0,1]$ because functions that admits a representation are  analytic functions on the unit disc. Actually, $f(t)=\sum_k a_k t^k$ immediately implies differentiability at $t=0,$ which is enough to conclude. So, it's more a matter of non-existence than non-unicity. 

With trigonometric polynomials it is a bit more delicate. But the reason is again non-existence of the uniform convergent expansion for some continuous function. 
Of course, if $f$ admits a representation as uniform limit of a series $\sum_k c_k e^{ikt}$ then the series is its Fourier series, so the claim that $\{ e^{ikt} \}_{k\in\mathbb{Z} }$ is not a Schauder basis for the continuous $2\pi$-periodic functions is equivalent to the statement that a Fourier series of a continuous $2\pi$-periodic function may fail to converge uniformly. (I do not have an example handy; one can also show it indirectly as a consequence of Uniform Boundedness principle if I remember well).