Are there infinitely many sets of relatively prime numbers with equal number and sum of divisors?

A quick computational note on (what I see as) the main question raised:

Do there exist infinitely many pairs of natural numbers $\{a,b\}$ such that $\tau(a)=\tau(b)$ and $\sigma(a)=\sigma(b)$ with the property that $\forall p\left( \nu_p(a)=\nu_p(b)\Rightarrow\nu_p(a)=\nu_p(b)=0\right)$?

I will be truly shocked if this answer is negative; simply by guessing and checking I have found quadruples of distinct primes $(p,q,r,s)$ such that $$ (p+1)(q+1)=24k=(r+1)(s+1) $$ for no less than $12$ values of $k$. I have listed them below:

$k=1\quad(3,5,2,7)\\k=2\quad(5,7,3,11)\\k=3\quad(5,11,3,17)\\k=4\quad(7,11,2,31)\\k=5\quad(5,19,3,29)\\k=6\quad(7,17,2,47)\\k=7\quad(11,13,3,41)\\k=8\quad(7,23,5,31)\\k=9\quad(11,17,3,53)\\k=10\quad(11,19,7,29)\\k=12\quad(11,23,5,47)\\k=14\quad(13,23,7,41)$

It seems to me that you are figuratively tripping over solutions to this problem. I would suggest trying to approach this question not through specifications into known conjectures like Dickson's Conjecture or Schinzel's Hypothesis, but rather by seeing how developed the literature is in solving Diophantine equations in the primes, like this paper does.