Because the Doctor and the Master both have habits when choosing aliases. Because why not? Because it's fun. I give you this fic. I do not own Doctor Who. Just assume that it's some mission between Eaters of Light and World Enough and Time.
Missy didn't think her day could get any worse. First she learned that concealing poison somehow counted as carrying weapons in the Doctor's eyes (something that she had been explicitly banned from doing the moment she got on board the TARDIS) and then she had to go on some ridiculous stealth mission (something about a corrupt organization to expose and/or take down, she wasn't paying much attention). And then the Doctor had to make it even worse.
She was having a perfectly good time at first, thinking of entertaining ways to annoy the Doctor during the mission (cover stories were always hilarious to mess around with or add embarrassing details to), and then they hit security.
"Names and intent of visit," said the security officer in a flat tone that clearly stated she'd gone too long without a vacation.
"Of course," the Doctor handed her his psychic paper, "I'm Doctor John Smith and this is my associate, Mistress Melissa Masters."
It took Missy a considerable amount of effort to avoid decking the Doctor then and there for sticking her with such a horribly obvious alias, but she suspected that was a bit of a no-no for covert operations, so she waited until they were past security before dragging him into the first little nook they came across.
"Did you seriously just stick me with that!?"
"What?" the Doctor asked, putting on the innocent look that really worked a lot better before this incarnation, "You do it yourself!"
"Well, I'm subtle about it, at least!" she countered.
The Doctor put on a thoughtful look. "Oh, really? Off the top of my head, your aliases have included Masters– that's obvious, Magister– Latin for 'Master,' Thascalos– Greek for 'Master,' Tremas– an anagram of 'Master,' need I continue?"
"Yes, yes. I get it, but my goodness; that alliteration was enough to give me a headache. Couldn't you have left me with something that won't stand out?"
"I thought you enjoyed standing out."
"Not in covert situations!" By then it was as close to screaming as either of them could get while maintaining a whisper.
"Ah," the Doctor hissed sarcastically, "And that's why you've been wearing period clothes and enough makeup to put a drag queen to shame."
Missy rolled her eyes, "As if your fashion choice wasn't the most tragic thing I've ever seen. You dressed like a gay vampire for years before replacing it with that horrid scarf and fedora. Oh! And let's not forget the clown coat. Seriously, how did your eyes not burn? And let's not forget that one time you wore a–"
"Okay," the Doctor interrupted, shifting to the defensive, "I admit, that coat was horrible. Now can we please get back to our task?"
Missy grinned smugly, glad to see she could still get the edge when she wanted to watch him squirm. "All right," she said, already working on new ways to exact revenge on the Doctor for her horrid alias.