It may be the same legal company, but that company lay dormant for a number of years. The former Cityflyer Express operation was merged into LGW short-haul. The RJ100s were shipped out to British Airways' regional out-stations at MAN and BHX, allowing their respective B737/A319 fleets to move down to LGW/LHR. So after the takeover Cityflyer Express ceased to exist as an operating company, and was merged into BA mainline at LGW in 2001.
When BA Connect, the former Manx/Brymon/Loganair/BRAL/BA CitiExpress operation was sold to flybe in 2007 the fairly recent LCY operation was the only part of the operation not included in the sale. However flybe did not need BA Connects Manchester head office. So the Manchester head office of BA Connect together with the EDI/LCY bases crews formed the genesis of the new BA Cityflyer AOC.
So BA used the legal vehicle of the former Cityflyer Express to form the new airline in 2007, as most staff/aircraft/routes of BA Connect became part of flybe.
That said, the management, offices and staff were from BA Connect. And while some of the aircraft may have been operated by the old Cityflyer Express early in their lives, they had all been part of BA Connect for 6 years before the new LCY-based BA Cityflyer came into existence.
There is no direct link between BA Cityflyer and Cityflyer Express in terms of company staff, infrastructure or aircraft. Using the old Cityflyer Express no doubt saved some money over setting up a brand new legal entity - the old Cityflyer Express had been dormant for 6 years when BA Cityflyer was formed out of the remnants of the BA Connect operation.