Talk to your coworker
Politely, without being sarcastic or testy, tell your coworker that it is your mug, and you'd like to use it - point out that it is moved by the cleaner, perhaps, and that you intend to store it on your desk to avoid causing problems.
If that seems fussy, well so be it - be fussy, or don't be fussy, but don't be fussy and try to hide it, or make it some unsaid grievance. The essence of this seems to be that you don't like what they're doing but you can't quite bring yourself to acknowledge the pettiness of the behaviour. If you think it is that petty, why do you care so much?
The problem with silently asserting ownership
I'm slightly troubled by many of the existing answers; they are either evasive (silently hoard the cup), passive aggressive (lock mug, writing your name on the mug, add bitterant(!!)) or outright aggressive (take mug from colleague, pour contents down sink). Aggression (passive or active) and evasion are not constructive ways to behave in a community like the workplace. If you are passive aggressive, and it happens again, next time you'll be angry, and quite possibly with no one knowing why.
Expectations in society
This is all about expectations. I'm not saying mugs should be communal, I'm pointing out that it is very evident that mugs are assumed to be communal at this workplace. The OP acknowledges that, and that it will seem petty to point out that he/she owns the mug. In that context, it is not about the facts of possession, but about how you establish that you are going to break with the general rule.
Of course if you own something you own it, but the context is that the coworker sees 1 mug in a collection of many that he/she safely presumes to be communal. You have to look at it from the coworker's perspective, and presume innocence.
Society in general functions on the basis of expectations, and different cultures have different expectations. For example, in the UK driving is very rule-bound, and you expect others to obey the rules. My colleague tells me that in his country, people ignore most of the rules, so you don't expect others to obey the rules. Neither is universally the 'right' system, but it is important that you know which you are in and make it clear to others if you are about to go against expectations. It is more dangerous to break the rules of the road when everyone expects you to obey them, because they won't be looking for people who do that.
Here, the expectation is that cups in the kitchen are communal, that you are free to choose a cup and use it. In that context, someone having their own mug is not wrong but it is different, so all they need to do is communicate that with the fellow members of the community.
Don't go through life thinking that your rules are the only right ones, and that everyone else should stick to your system, or you will rub completely normal people up the wrong way and make difficulties for yourself.
Is it that hard to talk to someone about a mug?