This question is a follow up to this excellent mathematics stackexchange question.
Let $\mu(n)$ be the Möbius function, $\phi(n)$ Euler's totient function, $\sigma(n)$ the sum of divisors function and $\tau(n)$ the number of divisors function. Define the set $S_N,$ for a natural number $N,$ by
$$S_N = \lbrace (m,n) \in \mathbb{N} \times \mathbb{N} \mid m \ne n, \, \mu(m)=\mu(n), \, \phi(m)=\phi(n),$$ $$\sigma(m)=\sigma(n), \, \tau(m)=\tau(n) \textrm{ and } \text{max} \lbrace m,n \rbrace \le N \rbrace .$$
How large is the set $ S_N $ ?