This is also found with the name Kerria japonica 'Flore Pleno'. Is one of them really a trademark name? I've only seen them both shown as the cultivar name, with a comment that they're synonyms.
Neither are trademarks. 'Pleniflora' is correct: Search for plants in the RHS Horticultural Database / RHS Gardening
Wikipedia tells us William Kerr introduced the cultivar 'Pleniflora'. Flore-pleno is Latin for double-flowered. Most of the sites I look at indicate the correct cultivar name as 'Pleniflora' with 'Flora Pleno' listed as the synonym. Stearn's dictionary of Plant Names indicates that in current usage the term flore-pleno is usually replaced by pleniflorus, duplex or plenus.