In the proof of Dirichlet's theorem, we show that $L(s,\chi_0)$ has a simple pole at $s=1$ where $\chi_0$ is the principal character and that $L(1,\chi)\neq 0$ otherwise. Therefore the logarithmic pole at $s=1$ of $\log L(s,\chi_0)$ is not canceled by a (negative) logarithmic pole of $\log L(s,\chi)$ at $s=1$ for any other $\chi$, so that the main term of the asymptotic expansion is $\frac x{\log x}$ multiplied by the residue of the pole.
What I don't understand is the following: in the proof of the prime number theorem, it is necessary to show not only the existence of a simple pole at $s=1$ and so a logarithmic pole of $\log \zeta(s)$ (in the case of the PNT we are concerned only with the Riemann zeta function, so there are no other L-functions to cancel the pole) but also the lack of any zeros, and thus (negative) logarithmic poles with real part $1$. Why is this not necessary in the case of Dirichlet's theorem?
To prove only the statement that there are infinitely primes in arithmetic progressions, you only look at the limit $$ \lim_{s \to 1} \sum_{p \equiv a \pmod n} \frac{1}{p^s} \tag{1}$$ and show that it's infinite. So the only point of analytic interest is $s = 1$. To understand $(1)$, one ends up looking at a sum of the Dirichlet $L$-functions $L(s, \chi)$ of with characters $\chi$ mod $N$.
A parallel statement can be said about the Riemann $\zeta$ function. Since $$\zeta(s) = \prod_p \left( 1 - \frac{1}{p^s}\right)^{-1},$$ and since we know $$ \lim_{s \to 1} \zeta(s) = \infty,$$ we know that there must appear infinitely many terms in the product. And thus there are infinitely many primes.
But the Prime Number Theorem (PNT) is much stronger than saying that there are infinitely many primes. It describes asymptotics of the number of primes and is obtained by performing a particular integral transform (an inverse Mellin Transform, or a Laplace Transform, or a particular line integral) and using more than merely local analytic behaviour.
If you want to prove the analytic asymptotics for primes in arithmetic progressions, then you can use very similar techniques. And for these, you do need to understand the analytic behaviour of each $L$-function on the line $\Re s = 1$.