This is an excellent question, which was the cause of some confusion in the field over the years:
"The validity of the second Chargaff rule was unexpected. Obviously it should be regarded as a global rule, i.e. applicable to large sections of chromosomes. Nonetheless, not being derived from a compelling principle, such as the one underlying the first rule, it remains a mystery. This is even more so, when one studies extended versions of Chargaff’s second rule." (paper)
There are a few things to note here:
- Chargaff's rules were purely observational; they pre-existed the double helix (see your same Wikipedia ref), and thus pre-existed the double helix model (which was in fact famously a rendering of the first rule in a 3-D model, as Watson wrote about in his book "The Double Helix"). There is thus no logical or mechanistic justification for the rules; they were simply what Chargaff et al. observed for dsDNA and ssDNA. Perhaps I have misinterpreted, but I felt the question implied a belief that there must be some logical requirement for this.
- Note that the first rule uses "=" and the second rule uses "~" ("approximately"). That is, if we believe that all bases in a genome are paired, then it follows as a mechanistic/logical requirement that Watson strand and Crick strand must match up compositionally. This is obviously not the case with ssDNA, as I think you indicate in the Q. However, it is nonetheless true that genomes have overall compositional biases; they tend to have consistent (G+C)/(A+T) ratios across the molecule (on average). More recently, it has become clear that this is true not only for single nucleotides, but also for $k$-mers, or short DNA sequences of length $k$ (paper). Indeed, these patterns have been used to match up different pieces from the same genomes (paper). These biases exist on both strands, therefore each strand of a genome is subject to these biases. (The reasons why these biases exist are often argued over, but may have something to do with mutational biases). Therefore, both ssDNA strands of a dsDNA genome have the same compositional biases, and they must complement each other's similar biases, so even within a strand the statistical expectation is Chargaff's second rule. Chargaff, again, probably didn't know many of these things when he came up with the rules. This explanation for Chargaff's second rule has been explicitly discussed in the literature, in the first paper cited.
Notably, the second rule only holds reliably for long sequences, because in short sequences sampling error may lead to a failure to match the biases of forward and reverse strands. This is more or less what everyone means when they call it a "global" rule.