You know that
$$f(x) = \frac{a_0}{2} + \sum_{n = 1}^\infty (a_n \cos(nx) + b_n\sin(nx)), \quad x \in [-\pi,\pi].$$
So
$$\int_{-\pi}^\pi f(x)g(x)\, dx = \frac{a_0}{2}\int_{-\pi}^\pi g(x)\, dx + \sum_{n = 1}^\infty \left(a_n \int_{-\pi}^\pi \cos(nx)g(x)\, dx + b_n \int_{-\pi}^\pi \sin(nx)g(x)\, dx\right).$$
This step is justified by the uniform convergence assumptions given in the problem. By definition of the Fourier coefficients $\alpha_n$ and $\beta_n$, we have
$$\pi \alpha_0 = \int_{-\pi}^\pi g(x)\, dx,$$
$$\pi \alpha_n = \int_{-\pi}^\pi \cos(nx)g(x)\, dx,\quad n \ge 1$$
and
$$\pi \beta_n = \int_{-\pi}^\pi \sin(nx)g(x)\, dx,\quad n \ge 1.$$
Therefore
$$\int_{-\pi}^\pi f(x)g(x)\, dx = \frac{\pi a_0\alpha_0}{2} + \sum_{n = 1}^\infty (\pi a_n\alpha_n + \pi b_n \beta_n),$$
or
$$\int_{-\pi}^\pi f(x)g(x)\, dx = \frac{a_0 \alpha_0}{2} + \sum_{n = 1}^\infty (a_n \alpha_n + b_n \beta_n).$$
The easy part is to show that $f$ and $g$ have the same Fourier coefficients. The hard part is to show that $f$ and $g$ are identically equal.
First show that $f$ and $g$ have the same Fourier series.
From periodicity of the series, we can consider any interval $[\alpha, \alpha+ 2\pi]$, and adopting standard notation replace $a_0$ with $a_0/2$.
We have
$$\tag{1}g(x) = a_0/2 + \sum_{n=1}^\infty (a_n \cos nx + b_n \sin nx)$$
where the series is uniformly convergent on the interval, and, since each term in the series is continuous, it follows from uniform convergence that $g$ is continuous.
Multiplying by $\sin mx$ and integrating we get
$$\tag{2}\frac{1}{\pi} \int_{\alpha}^{\alpha + 2\pi}g(x) \sin mx \,dx \\= \frac{a_0}{2\pi}\int_{\alpha}^{\alpha + 2\pi}\sin mx \, dx +\frac{1}{\pi}\int_{\alpha}^{\alpha + 2\pi}\sum_{n=1}^\infty(a_n \cos nx \sin mx + b_n \sin nx \sin mx) \, dx.$$
Since (1) is uniformly convergent and $\sin mx$ is bounded, the series on the RHS of (2) is uniformly convergent and can be integrated term by term to obtain
$$\tag{3}\frac{1}{\pi} \int_{\alpha}^{\alpha + 2\pi}g(x) \sin mx \,dx \\= \frac{a_0}{2\pi}\int_{\alpha}^{\alpha + 2\pi}\sin mx \, dx +\sum_{n=1}^\infty\frac{a_n}{\pi}\left(\int_{\alpha}^{\alpha + 2\pi}\cos nx \sin mx \, dx \right) + \frac{b_n}{\pi} \left(\int_{\alpha}^{\alpha + 2\pi} \sin nx \sin mx \, dx \right)$$
All the integrals on the RHS of (3) vanish except the integral of $\sin nx \sin mx$ when $n = m$.
Hence,
$$\frac{1}{\pi} \int_{\alpha}^{\alpha + 2\pi}g(x) \sin mx \,dx = b_m = \frac{1}{\pi} \int_{\alpha}^{\alpha + 2\pi}f(x) \sin mx \,dx $$.
Similarly we can show
$$\frac{1}{\pi} \int_{\alpha}^{\alpha + 2\pi}g(x)\,dx = a_0 = \frac{1}{\pi} \int_{\alpha}^{\alpha + 2\pi}f(x)\,dx , \\ \frac{1}{\pi} \int_{\alpha}^{\alpha + 2\pi}g(x) \cos mx \,dx = a_m = \frac{1}{\pi} \int_{\alpha}^{\alpha + 2\pi}f(x) \cos mx \,dx. $$
Thus $f$ and $g$ have the same Fourier series.
It still remains to prove that $f = g$.
This follows by showing that if two continuous functions differ at just one point, $f(c) \neq g(c)$, then they cannot have the same Fourier series.
Take $h = f -g$. If $f$ and $g$ have identical Fourier series, then all Fourier coefficients of $h$ vanish, and for any trigonometric polynomial $T_m(x) = A_0/2 + \sum_{n=1}^{m} (A_n \cos nx + B_n \sin nx)$ we have for any $\alpha \in \mathbb{R}$,
$$\tag{4}\int_{\alpha}^{\alpha + 2\pi} h(x) T_m(x) \, dx = 0.$$
We also have $h(c) = f(c) - g(c) \neq 0$ and WLOG can assume that $h(c) > 0$. Since $h$ is continuous , there exists $K > 0$ and $\delta > 0$ such that $h(x) \geqslant K > 0$ when $x \in [c - \delta,c + \delta]$.
It can be shown that $T_m(x) = [1 + \cos(x-c) - \cos \delta]^m$ is a trigonometric polynomial satisfying
$$T_m(x) \geqslant 1 \text{ for } x \in [c-\delta,c+\delta] \\ \lim_{m \to \infty} T_m(x) = \infty \text{ uniformly on } [c - \delta/2, c+\delta/2] \\ |T_m(x)| \leqslant 1 \text{ for } x \in [c + \delta , c - \delta + 2\pi]
$$
From these properties it follows that
$$\tag{5} \int_{c - \delta}^{c + \delta} h(x) T_m(x) \, dx \geqslant \int_{c - \delta/2}^{c + \delta/2} h(x) T_m(x) \, dx \geqslant \delta \, K \, \inf T_m(x) $$
and $\int_{c + \delta}^{c - \delta + 2\pi} h(x) T_m(x) \, dx$ is bounded.
Since the RHS of (5) tends to $\infty$ as $m \to \infty$ it follows for sufficiently large $m$
$$\int_{c - \delta}^{c - \delta + 2\pi} h(x) T_m(x) \, dx \neq 0,$$
contradicting (4) and leading to the conclusion that if $f$ and $g$ differ at one point, they cannot have the same Fourier series.
Best Answer
Suppose $$ f(x)= \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n\cos(nx) + b_n\sin(nx) $$ uniformly on $[0,2\pi]$. Then, $$ \int_0^{2\pi}f(x)\cos x\,dx =\int_{0}^{2\pi} \left(\frac{a_0}{2} + \sum_{n=1}^{\infty} a_n\cos(nx) + b_n\sin(nx)\right)\cos x\,dx. $$ But "uniform" convergence allows you the switch the integral and summation signs, which implies that $$ \int_0^{2\pi}f(x)\cos x\,dx=a_1\int_{0}^{2\pi}\cos^2x\,dx=a_1\pi. $$ With the exactly same argument, $$ \int_0^{2\pi}f(x)\cos x\,dx=c_1\int_{0}^{2\pi}\cos^2x\,dx=c_1\pi. $$ This tells you that $a_1=c_1$.
Similarly, you can show that $a_n=c_n$, by multiplying $f$ with $\cos(nx)$, and $b_n=d_n$, by multiplying $f$ with $\sin(nx)$.