Can the " same base, same exponent " rule be extended to sums?

A perhaps more worrisome example is that $$ (-1)^a = (-1)^b $$ has infinitely many solutions in integers. The equation holds if $a$ and $b$ are both even and it holds if $a$ and $b$ are both odd. This is very far from requiring $a = b$.

No.

It's better: $$(2^x)^2+2^x-6=0$$ or $$(2^x-2)(2^x+3)=0$$ or $$2^x=2,$$ which gives $$x=1.$$ Also, you can say that $2^{2x}+2^x$ increases, which by your work gives $x=1$ again.

If one of the bases is less than one, you can get, for example $$0.5^x+2^x =2.5$$ which has two solutions $x=1$ and $x=-1$