By chance, I have the same last name as one of the supervisors of a project I worked on. If I ask him to write me a recommendation letter, will the admission committee misconstrue that we are related and how do I provide proof that we are not? By the way, I am Chinese, so there is perhaps more overlap in last names.
There is a very easy solution. Your supervisor can simply write something like
To whom it may concern,
It is a pleasure to write in recommendation of John Smith (no relation). John is...etc., etc.
Professor Tom Smith
By the way, I am Chinese, so there is perhaps more overlap in last names.
Perhaps? Come on. Any worldly person who is paying attention knows that the ratio of people to surnames is orders of magnitude higher in China than in most Western countries. There are more than 1.3 billion Chinese (i.e., citizens of the PRC) and only slightly over 4,000 Chinese surnames, 100 of which account for about 85% of China's population. A 2007 survey reports approximately 92,881,000 Chinese with the surname Wang: more than 7 percent of the population. (Only about 1% of Americans have the surname Smith, and it drops off much more rapidly from there.)
I completely agree with @Corvus's answer: with a two word parenthetical expression any writer can allay all concerns in this regard, and it is probably a good idea to do so. (When I describe work of other mathematicians named "Clark" -- and, because I am ever tolerant of the imperfections of others, "Clarke" -- I generally do say "no relation".) But I wouldn't worry too much about this. You're Chinese, not Macedonian: even we Americans know a thing or two about where you're from.