Following the convention of the Royal Horticultural Society, I’ve used the name Rubus Tayberry Group for this hybrid. The original tayberry, a cross between the early-ripening Rubus ‘Aurora’ (a blackberry) and a large-fruited tetraploid raspberry (tetraploids have 4 sets of chromosomes instead of the usual two and tend to be more robust), was hybridized at the Scottish Crops Research Institute in Invergowrie, Scotland (source: Wikipedia).
In efforts to improve it (e.g., a longer fruiting season or improving hardiness or thornlessness), other plant breeders would attempt similar crosses but with slight variations (e.g., a different blackberry cultivar as one parent) — the end results would be something closely resembling the original tayberry, but genetically different. Since a cultivar name should, in theory, represent only one distinct genotype, a different naming convention is required when an aggregate of (often) closely-related genotypes are known to exist, though it may be exceedingly difficult to tell them apart. Hence, Tayberry Group. Similar examples include Colorado blue spruce (Picea pungens Glauca Group), purple New Zealand flax (Phormium tenax Purpureum Group) and pendulous ginkgo (Ginkgo biloba Pendula Group).
The flavour of tayberry has been described as “clean, tangy, and tart”. I find them to have a slight overtone of soap which isn’t altogether pleasant, but perhaps that’s just me.