Scientific theories and falsification are not generally questions about specific instances, but applied to general processes and statements about nature. Theories make predictions, then you perform experiments to see if the results are consistent with the theory's predictions (adding support to the theory) or inconsistent (falsifying it).
But for specific instances, the "scientific method" above doesn't really apply. For some random thing you find, a scientist would start by trying to classify it among known objects that it's similar to. If the processes that create these objects are known, the initial assumption would be that it was created using similar processes -- the "if it walks like a duck" method. So if you find a clock, and you know that clock-making is something that people do, it's most likely that it was created by a clockmaker.
Conversely, if you find a living thing, the obvious assumption would be that it was created by evolution through natural and/or directed selection, since we know that this is the usual process that produces new organisms.
Once you make that initial assumption, you can then examine the object more closely, to see if you can determine more specifically how the creation process applies.
If the object doesn't bear any resemblance to things that we already know about, or you can't figure out the specifics, we have a mystery. If an existing theory comes close to explaining it, we may need to make adjustments to that theory to accomodate it; for instance, once we discovered the genetic mechanisms underlying evolution, there were tweeks to accomodate genes jumping between species.
If not that, we need to do more investigation. Hopefully we can eventually find more instances of this new object (and the same thing applies to phenomena). With more examples, we try to find some commonality (essentially creating a new class of objects), and form hypotheses about this class. Only then can we make predictions about the class that can be confirmed or falsified.
Of course, statements about an individual object can be true or false, but it isn't always possible to prove all such statements. The claim "X was designed" can be shown to be true if you can find the designer. But as the saying goes, you can't prove a negative: if you can't find the designer, you can't be absolutely sure. But if you know natural processes that commonly produce the object, it's not unreasonable to assume that it was not designed until something contradicts that assumption.
As a simpler thought experiment, consider Nelson's "grue", an object that's green before time t and blue after t. If you encounter a green object before t you can wait until t and see if it becomes blue. But if you encounter a blue object after t, you can't determine whether it was green before t. (And both these statements ignore the infinite times surrounding the observations, for simplification purposes.) However, if there are other objects that have been determined to be grue, and this object shares many properties with it, we may naturally assume it's also grue (similar to the way we conclude a creature is a dog or cat based on superficial observations).
We can't always be sure of everything. There are still plenty of holes in our knowledge. Science is a process of progressively filling in those holes, although it also reveals new holes. For instance, as we discovered more details about cosmology, we learned that dark matter and dark energy exist, and now we're investigating what they're made of.
