Oversampling can be used to reduce noise only when you're oversampling the same quantity and the noise samples are uncorrelated and have zero mean. This is seldom the case.
Let's say you're sampling signal $s(t)=r(t)+n(t)$, where $r(t)$ is the signal you're interested in, and $n(t)$ is zero-mean noise, which comes primarily from internal noise in your circuitry. You oversample by a factor of 2 and then average every two samples. Your samples are, say, $s[1]$ and $s[2]$, taken at times $T_s$ and $2T_s$. The average is
$$\begin{align}
s_a&=\frac{s[1]+s[2]}{2} \\
&=\frac{r[1]+r[2]}{2}+n_a \\
&=r_a+n_a
\end{align}$$
where $n_a$ is, you hope, closer to zero than a regular noise sample (if the noise samples are correlated, their average is not necessarily close to 0). However, what can you say about $r_a$? It is neither $r[1]$ nor $r[2]$. You may hope it is close to $r(1.5T_s)$ (or something) but there is no guarantee of that.
Now consider this scenario: you feed $r(t)$ into two circuits, each with its own ADC, in parallel, with synchronized sampling clocks. At time $t=T_s$ you have two samples, $s_1[1]$ and $s_2[1]$, one from each converter, and (this is key) each with independent, uncorrelated noise. Now your average is
$$\begin{align}
s_a&=\frac{s_1[1]+s_2[1]}{2} \\
&=\frac{r[1]+r[1]}{2}+n_a \\
&=r[1]+n_a
\end{align}$$
Now you have achieved an actual reduction in noise. You achieved that by sampling the same quantity ($r[1]$) twice, and by making sure the noise samples are uncorrelated (since the noise is produced by two different circuits).
