Is $\limsup_{x\to\infty}\big(\sum\limits_{d|3^x-1}{1/d}\big)/\big(\sum\limits_{p<x}1/p\big)<\infty$?

The OP's first display is true. Indeed, Erdős (1971) proved that $$\sum_{d\mid 2^x-1}\frac{1}{d}\ll\log\log x\ll\sum_{p<x}\frac{1}{p},$$ and he noted that the proof works equally well for the divisors of $a^x-1$.

One remark for the OP. On certain well-believed conjectures, the exact value of the limsup can be determined. The answer involves $\gamma$, $\log 3$, and an integral expression with Dickman's rho function. See this arXiv paper: https://arxiv.org/abs/2006.02373 (Theorem 1.2 and section 6)