You are right to question this. The product $\prod_p \left(1 - \chi(p)/p\right)^{-1}$ (where $\chi = (-1/\cdot)$ is the Dirichlet character mod $4$) does converge, but that requires justification; indeed it is equivalent to the non-vanishing of the Dirichlet function $L(s,\chi)$ on the edge $s = 1+it$ of the critical strip, which is also what you need to prove the analogue of the Prime Number Theorem for primes in arithmetic progressions mod $4$. (Taking logarithms we see that the desired convergence is equivalent to convergence of the sum $\sum_p \chi(p)/p$, which differs from the product's logarithm by an absolutely convergent sum $\sum_p O(1/p^2)$, and from that to $L(s,\chi)$ is classical analytic number theory.)

You are right to question this. The product $\prod_p \left(1 - \chi(p)/p\right)^{-1}$ (where $\chi = (-1/\cdot)$ is the Dirichlet character mod $4$) does converge, and the limit is $L(1,\chi) = \pi/4$ as expected; But this requires justification $-$ indeed it is equivalent to the non-vanishing of the Dirichlet function $L(s,\chi)$ on the edge $s = 1+it$ of the critical strip, which is also what you need to prove the analogue of the Prime Number Theorem for primes in arithmetic progressions mod $4$. (Taking logarithms, we see that $\prod_p \left(1 - \chi(p)/p\right)^{-1}$ converges if and only if $\sum_p \chi(p)/p$ converges, since this sum differs from the product's logarithm by an absolutely convergent sum $\sum_p O(1/p^2)$; getting from $\sum_p \chi(p)/p$ to $L(s,\chi)$, and then showing that the product $\prod_p \left(1 - \chi(p)/p\right)^{-1}$ actually converges to $L(1,\chi)$, is a classical chapter of analytic number theory.)