Let's solve the equation step by step:
The given equation is:
\[ \frac{6x - 6}{3} = 2x - 2 \]
First, simplify the left side:
\[ \frac{6x}{3} - \frac{6}{3} = 2x - 2 \] \[ 2x - 2 = 2x - 2 \]
Now, we can see that the equation simplifies to:
\[ 2x - 2 = 2x - 2 \]
This means that both sides of the equation are identical. This implies that the equation is true for all values of \( x \).
Since there are no restrictions, we conclude that there are infinitely many solutions.
Thus, the correct answer is:
D: infinitely many solutions.