The rescalling $(V')$ of $(V)$ as described in the question has always an isochronous center when $(V)$ is a quadratic system with center.

The reason is that  $d\theta(V')=1$ where $d\theta=(\frac{1}{x^2+y^2})(ydx-xdy)$.


**Remark:** The  same  argument as  above  can be  applied to  show  the following:

>Let $V$ be  an arbitrary quadratic  vector  field on the  plane and $(V')$ be  the  corresponding rescalling as in the  question. (We no longer  assume  that $V$  has necessarily   a  center).  Then  all  closed orbits or  limit  cycles  of  $V'$ which surround the  origin have  the  same  length provided we choose a  Riemannian metric whose frame  is  in the  form  $\{V', f(x\partial_x+y\partial_y)\}$  where  $f$  is  an  arbitrary  positive  smooth function.