Laura Kuenssberg is leaving the BBC to become the Business Editor of ITV, leading to speculation that her Twitter handle @BBCLauraK (with its 58,832 followers) may have to be changed to @ITVLauraK. I even wonder whether some enterprising soul at the independent broadcaster might have already asked the BBC how much they want for that list or whether some coding genius is working on a way that old Twitter names could be subsumed into new ones.
In the course of a long and no doubt distinguished career somebody like Kuenssberg can expect to work for many different organisations (as well quite a few that remain the same and yet change their names) so it doesn't make any sense for her to sell or lease her identity to them. It's enough to sell or lease her services.
On a less exalted level I've always told my kids that when they start work they should use their full name in every interaction. There's no use developing strong recognition as "Jane from Acme Magazines" because it's obvious that one day you won't be that any more and you'll have to start all over again, identifying yourself with some other organisation. Build your own brand. It'll last longer than theirs.