In the first picture, in the background is a piece of wood with an "antiqued" type of glazed finish. Is that what the finish is on the rest of the piece? If so the black dots on your wood could be "fly specks" from when they finished the rest of the piece. I don't think they belong in the inside of your drawers so unless you like them I would take it back.
It doesn't look like mold to me. If it is flyspeck that should easily sand off. I think more likely the spots are some insect damage to the tree as it was growing and could just be ignored unless you are put off by the appearance.
If it's mildew, it will wipe off with a little chlorine bleach mixed with water. Fly specking is a technique used in finishing. A black or brownish paint is flicked/splattered on the surface for visual effect. Real fly specking is fly manure, you can feel it as small bumps on the surface and it's not distributed evenly.
I don't think that it's mold. Why? If it was, the infection would have had a more synchronous start and the speckles/freckles would have been all of a similar size, under nearly equal growth conditions. Then it would look like a lot of the birch that I buy!
I suggest it's a really clumsy overspray from an adjacent finishing project.
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