Is the birch tree a fruit bearing tree?

Yes, it is. Most people misunderstand what fruit is; Apple, peach, banana, plum, pear, those are obvious fruit trees that everyone would recognize as such, but this is going by the purely horticultural definition of fruit.

But more broadly, all trees that produce flowers are fruit-bearing trees, because the seed-containing ovaries are classed as fruit. So yes, the birch tree is a fruit-bearing tree.

An interesting note about birch is that its fruit was used as a food source by the Incas.