Eve isn’t her real name. It’s just a little joke she plays with herself. And the government of course, but she is always playing with them. (It’s a matter of temptation) Eve is betrayal, but Eve is also awareness and knowledge. Knowledge of others and who they are and want they want, but more importantly knowledge of herself. She knows precisely who she is. Not many of the people who work here can say that.
Moneypenny on the other hand is her real name. It is a ridiculous name in her opinion and she would in many ways quite like to be rid of it. Shake herself free of the last pieces of herself which are not all smooth lines and high heels and a stab to the gut. But it has its uses. She hasn’t kept something that ties her to her old life out of any sense of sentimentality.
She kept it because it starts with the letter M.
And MI6 seem to have a disturbing tradition of appointing people based on their names, rather than their qualifications. After all, Mallory took a demotion to get his current job.
She knows how they work, even if she thinks it’s patently ridiculous.
(It’s one of the first things she plans to change. It’s at the same level of ridiculous as appointing someone based on their title. They certainly don’t live in the 19th century anymore.)
Until then though, the name will stay. In her opinion, useful things can always stick around, regardless of how irritating they might be otherwise.