When adding polynomial features, the issue of multicollinearity doesn't hold. Why? In the regression model, sometimes to capture the non-linear relationship between dependent and independent variables, we use polynomial features. But in regression, if the two or more features are correlated with each other, then regression coefficients become unstable, and non-interpretable, Causing the standard error of the model unreliable. But such a problem doesn't occur when we use polynomial features. Why is this the case?

In a regression model
$$E[Y|x,z]=\alpha_0+\alpha_1x+\alpha_2z$$
people often want to interpret $\alpha_1$ as some sort of effect of $x$ holding $z$ constant, or less ambitiously, a contrast for observations different in $x$ but the same in $z$.  We thus care about conditions under which those interpretations hold, and also under which
$$E[Y|x]=\beta_0+\beta_1x$$
has $\beta_1\approx\alpha_1$ and so on. We might also care about how the Fisher information for $\beta_1$ and $\alpha_1$ compare -- how much harder is the estimation because of collinearity.
For a model
$$E[Y|x]=\alpha_0+\alpha_1x+\alpha_2x^2$$
people don't usually want to talk about varying $x$ while holding $x^2$ constant, because that makes no sense.
So, the sorts of problematic statements people want to make about models with correlated predictors aren't a problem for polynomials because people are much less likely to want to make those statements.
As a separate issue, extreme collinearity, where the design matrix is nearly numerically singular, is as much or more of a problem for polynomials as for distinct predictors. It's just that it's pretty unusual now that everyone works in double precision and can lose ten digits of $\hat\alpha$ to rounding error without hardly noticing.
