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It's a statistical argument. That the possibility space that exists for snowflakes when forming is so vast that it is unlikely that two identical snowflakes have ever, or will ever, be created. Now I'm no mathemagician or physiwizard, but it seems to me that given the relative size of atoms in any structure that we can see with the naked eye makes this a pretty solid assertion. If we're including microscopic snowflakes, the space shrinks and at some point the argument falls apart, but I think we can all agree that a snowflake is a macroscopic object for the sake of this discussion.
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Yeah it's impossible to prove a universal claim like this.
But we can analyze water crystal formation and apply statistics and discover a rational belief. >

It's also weird that snowflakes get such emphasis. You could say the same about trees, or grains of sand, or pretty much any natural object >

Snowflakes are much prettier than those things.  Flowers could work, but then again 2 sunflowers look pretty identical often while snowflakes have distinctly different geometry. >

How "wide" is your definition of unique. 
Seems reasonable to suggest an identical shape with the same molecules is incredibly unlikely. >

Technically speaking, every snowflake is unique in the sense that the atoms that make up one snowflake are completely different from every other one to ever exist.  But as far as appearance goes, that might produce a different answer. >

As far as appearance goes, some might look the same under a microscope but then if you get a more powerful microscope, you may see a difference >

It actualy can be proven, cause the format is intrinsicaly related to temperature, humit and pressure, forming a caotic sistem. Since the number of arrangements is almoust infinite and the process that creats it is almost random, the probability of two identical ones emerging is disconciderable. It's like the formation of digitals, almost infinite combinations and a caotic process of creation assures that it is unique. >

I understand and agree that the odds of two snowflakes being exactly the same are extremely low, however I don't think theoretical proof is the same as actual proof.
I'm reminded of that old theory that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will eventually type the complete works of William Shakespeare. Again, a theory and not proven, but wouldn't the same theory hold true for snowflakes? With infinite time, two identical snowflakes could eventually form. Maybe one of them is in Siberia in the year 642 BC and one of them is in Colorado in 2027 - it'd be impossible to prove that they're identical but they still could be. >

the thing is that we don't have infinite time. go see this coment on r/mathmemes >

Sure, it may not be proven, but we can enjoy the beauty in each and every snowflake's uniqueness! >

Errr.... There's pictures of snow under a microscope 🔬 ya know? >

How does that prove they're all unique though? >

Today OP learned about underdetermination. >

There was that one guy who took pictures of thousands of snowflakes and not a single one of them looked alike