Ole, a few words in a comment below about the pseudoautosomal regions. Those are the only areas where recombination between the X and Y happen. Under the (outdated) genome map version GRCh37/hg37 that all our testing and reporting companies still use, on the X-chromosome PAR1 starts at locus 60,001 and ends at 2,699,520. PAR2 starts at 154,931,044 and ends at 155,260,560.
That segment you found at 1,370,495 to 88,78,959 would be partially in PAR1, but only about 1.33Mbp of it. So the portion that could have been affected by recombination would be very small; using the sex-averaged values our testing/reporting companies give us, at most a fraction over 3cM.
What company is reporting that 19.6cM segment? If GEDmatch, they don't, to my knowledge, actively screen out any PAR SNPs they get from the uploaded raw data, but they don't do match reporting on them. If it's GEDmatch, my first-pass, starting position would be to assume that the segment on the X is functionally closer to 16cM to 17cM than 19.6.
Given the scenario you describe, I'd tend to think there is something amiss with that particular X-chromosome result, and especially so since the segment begins in the PAR. If it's on GEDmatch, can you try manually restricting the SNP density at comparison to 500, and then 700, to see what the outcome is? It's possible we could be seeing a false segment because a few small--and likely meaningless--ones are being "spliced" together because they're just barely within the mismatch threshold at the default floating 200-400 SNPs.
Since both parents are tested (same company at about the same time?), there should be no segment found in the daughter that isn't found in one of the parents.