Kindly see the green underline. Why didn't Stewart just commence with $y$ as a positive number? Why define $a$ as "a positive constant", then in the last line replace "a by any [positive] number y"?
I know that the domain of logarithms is {positive real numbers}.
Stewart, Calculus 7th ed 2011. Not Early Transcendentals. p. 422 scanned.
Lots of narratives in Stewart's book on calculus should be understood with context. That book is not a rigorous real analysis textbook. You would have many troubles if you read it seriously line by line. If you want to learn real analysis from scratch so that every single word has a precise meaning, you shall try another one.
Going back to your question, the symbol $y$ is mentioned in the red box as a positive real number. So the author may consider it clear to not mention it explicitly again.
Ultimately, the choice of symbol doesn't matter. It doesn't matter if you put $a$, or ${y}$, or ${☺}$. The author just changed symbol to try and get you to think a bit about the context.
To start with, $a$ was picked first and fixed. $a$ was picked first. Then we showed ${\ln(ax) = \ln(a) + \ln(x)}$ ($x$ is some varying number, we needed it to be a variable so we could use Calculus and take a derivative). The point is, $a$ was picked and fixed at the start.
But the choice of $a$ at the start was arbitrary. This means for all positive numbers ${a,x}$: ${\ln(ax) = \ln(a) + \ln(x)}$. The author chooses to replace $a$ with $y$ to try and symbolise the fact that this choice of $a$ was arbitrary, and could be anything. $y$ is more commonly used to mean some potentially varying quantity. There's no significance behind the change in symbol apart from this contextual one.
Hope that helped.
Indeed he could have phrased the argument like this:
"Let $y$ be a positive real number. Let $f$ be the function defined by $f(x) = \ln(xy)$. (Here $x$ can be any positive real number.) Notice that $f'(x) = \frac{1}{x}$. It follows that $f(x) = \ln(x) + C$ for some constant $C$. Plugging in $1$ for $x$ reveals that $C = \ln(y)$. Thus, $\ln(xy) = \ln(x) + \ln(y)$."
That does seem slightly simpler to me. Perhaps, pedagogically, Stewart thought the use of the letter $a$ would make it easier for students to compute the derivative of $\ln(ax)$, as it is more similar to notation used previously in the textbook.
