If one modernises Plato's language, there is a sense in which he believes that 'something can be considered fake just because it's man-made'. Relevant here is his theory of the Forms as set out in The Republic.
In Plato's view, reality is a realm of entities, the Forms (eide), which are not accessible by the senses but only by reason. Only the Forms are real: for instance, only the Form of a circle is really, perfectly circular. Anything which I, you or anyone else might create would not be absolutely, perfectly, unqualifiedly circular; it would be in some degree incompletely circular. However close it might come to perfect circularity, it would never actually reach it. Plato's reasons for this claim are a separate matter; there is not space to consider them here.
With this view in mind, it would not be too inaccurate to say (in our own language, not Plato's) that anything man-made is not 'the real thing'. The circle that I draw is a 'fake' circle in the sense that it is only an imperfect, imitation circle. The authentic or original circle is the Form of the circle.
Two points need to be added. In the first place, as we use the term,'fake', a fake and an original can be qualitatively indistinguishable. In the Platonic case, the
man-made circle cannot be qualitatively indistinguishable from the Form of the circle since any man-make circle is not, and the Form of the circle is, perfectly circular.
Secondly, since Plato never offers a complete enumeration of the Forms, we cannot be sure that for everything that is man-made there is a corresponding Form. But where there is such correspondence, the fakeness of the man-made, in the sense explained, holds good.