Relative consistency of slightly modified GCH

Is it consistent with ZFC that $2^\kappa=\kappa^{++}$ for all regular $\kappa$ and $2^\kappa=\kappa^{+}$ for all singular $\kappa$?

Sure.

Easton's theorem tells us that if $F(\kappa)$ satisfies $\operatorname{cf}(F(\kappa))>\kappa$, then it is consistent that for all regular cardinals $F(\kappa)=2^\kappa$, and for singular cardinals the least possible value is taken.

Starting with a model of $\sf GCH$ the function $F(\kappa)=\kappa^{++}$ satisfies the needed conditions, and it is not hard to check that for singular cardinals the least possible value is indeed $\kappa^+$.

So this theory is indeed equiconsistent with $\sf ZFC$. If, however, you want for all cardinals to satisfy $2^\kappa=\kappa^{++}$, you will have to assume the consistency of some large cardinals as this will imply the violation of $\sf SCH$ at every singular cardinal. But this is possible, if you assume the consistency of enough large cardinals. This MathOverflow thread and the linked questions there are relevant to this.