In my textbook from p 438 on it is explained, that for example for the Ginzburg-Landau model defined by the effective Lagrangian

\[\mathcal{L}_{eff} = \mathcal{L}_{kin} -\frac{1}{2}(T-T_C)\phi^2
-\frac{1}{4!}\lambda\phi^4 +\cdots\]

the position of the non-trivial (Wilson-Fisher) fixed point which is found in this case by an $\epsilon$ expansion and dimensional regularization

\[\lambda_* =\frac{16\pi^2\epsilon}{3}, \quad m^2_* = 0\]

depends on the RG scheme applied and is therefore not physical. Only the critical exponents (that describe the behavior of the RG flow near the fixed point) are universal.

Is it always the case that the position of fixed points (or even more generally the structure) of the RG flow field depends on the renormalization scheme applied?

Concerning the fixed points, I dont understand why their position (and maybe even their presence or absence?) should depend on the scheme, as I always thought that fixed points of the RG flow corresponds to a scale (or even conformal) invariant theory, their basins of attraction define universality classes etc, so they should be physical and not depend on the exact renormalization method (scheme) applied?