Is there a logarithm base for which the logarithm becomes an identity function?

For a function to be a logarithm, it should satisfy the law of logarithms: $\log ab = \log a + \log b$, for $a,b \gt 0$. If it were the identity function, this would become $ab = a + b$, which clearly is not always true.

Note that $$\log_b x=x\iff b^x=x\iff b=\sqrt[x]{x}.$$ Since $\sqrt[x]{x}$ is not a constant function, the relation cannot hold for all $x$.

But it can be true for some particular $x$. For example $b=\sqrt{2}$ and we have $$\log_{\sqrt{2}}2=2.$$

No, it can't. For any base $b$, there is some real constant $C$, s.t. $$ \log_b x = C \ln x $$ If it were that this logarithm is identity function, then natural logarithm would be just $x/C$, which is clearly false.