MWE:
Let us consider a polynomial :
\begin{equation}
f(x)=a_0+a_1x \tag{1}
\end{equation}
where integer coefficients $a_0,a_1\in [0,2^r-1], r\in \mathbb{N}$.
Let $y_1=f(1)\pmod{2^r}$ and $y_2=f(2)\pmod{2^r}$
By Lagrange interpolation polynomial,
\begin{align*}
\phi(x) & =\sum\limits_{i=1}^2 y_i \prod_{j=1, j\neq i}^{2}\dfrac{x-x_j}{x_i-x_j} \pmod {2^r}\\
&= c_0+c_1x.
\end{align*}
Are the polynomials $f(x)$ and $\phi(x)$ unique? If yes how can I prove this?
Generally polynomials are not uniquely factorable modulo composites. In your example, take $r = 2$, $a_0 = 0$, $a_1 = 2$, so $f(x) = 2x$ mod 4. This is a linear polynomial but it has two roots, at $x=0$ and $x=2$.
More specifically, the formula you give for Lagrange interpolation fails because it divides by $(x_i - x_j)$, but inverses don't always exist mod composites (for the case you've chosen, you'll be unable to divide by even numbers because they share a common factor with $2^r$).