I believe the formula $|A|=\frac{\varphi(q)}{\mathrm{lcm}(\varphi(d_1),\varphi(d_2))}$ is not quite correct. In particular, for $q=15$, $d_1 = 3$, $d_2=5$, $a=1$, this formula gives $|A|=2$, while, in fact, $A = \{ (1,1) \}$ with $|A|=1$. Below I derive a correct formula.
First, it is clear that if $a\not\equiv1\pmod{\gcd(d_1,d_2)}$, then $|A|=0$. For the rest assume that $a\equiv1\pmod{\gcd(d_1,d_2)}$.
Let $p$ be a prime dividing $q$ and $t=\nu_p(q)>0$ (i.e., $t$ is the valuation of $q$ w.r.t. $p$) and $s_1 = \nu_p(d_1)$, $s_2 = \nu_p(d_2)$. It is easy to see that the number of elements modulo $p^t$ in $A$ is indeed $$\frac{\varphi(p^t)}{\mathrm{lcm}(\varphi(p^{s_1}),\varphi(p^{s_2}))} = \begin{cases} (p-1)p^{t-1},& \text{if}\ s_1=s_2=0\\ p^{t-\max\{s_1,s_2\}},&\text{otherwise}. \end{cases}$$
Now, for any $a$ such that $\gcd(a,q)=1$ and $a\equiv1\pmod{\gcd(d_1,d_2)}$, we have (thanks to CRT) $$ |A| = \prod_{p|q} \frac{\varphi(p^{\nu_p(q)})}{\mathrm{lcm}(\varphi(p^{\nu_p(d_1)}),\varphi(p^{\nu_p(d_2)}))} = \frac{\varphi(q)}{\prod_{p|q} \mathrm{lcm}(\varphi(p^{\nu_p(d_1)}),\varphi(p^{\nu_p(d_2)}))}.$$ Notice that this product does not collapse into $\frac{\varphi(q)}{\mathrm{lcm}(\varphi(d_1),\varphi(d_2))}$.
