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.)