The backing of 'fabric-like' nappies is a synthetic non-woven sheet, typically made of the same plastic (polyethylene) as the smooth film of a traditional plastic-backed product. However as it is a non-woven rather than a film it is porous and has to be sealed or partially sealed with a barrier layer. This can be made either as impervious as the traditional plastic film, or it can genuinely be made breathable, e.g. by including calcium carbonate that forms tiny pores in the plastic, too small for bulk liquid to flow through. This was being done as early as the 1980s and I suspect that most high-end baby diapers still have this microporous liquid barrier.
Whether there is a genuine advantage in this is another matter. If babies could choose their own diapers, would they opt for the suppleness and breathability of the nonwoven, or the security and reliability of the traditional plastic film? Presumably market research has shown that parents think the non-woven is better for their little ones, although most parents don't have direct experience of the pros and cons of each, not being in diapers themselves. If they were, they might be less keen on the non-wovens and more willing to purchase a traditional plastic-backed alternative for their offspring, in which case they would still be available as they are for those of us who can vouch for its advantages through experience.