The table top is still sipping out sap. Is this even possible from something that is 160 years old?
Sure, it's definitely possible. Sap doesn't just disappear from a piece of wood because it's old.
One might think that over that long of a time, the sap should have dried up a little. If it was kept in a cooler place and brought into somewhere warmer, the sap could start to flow again.
made from reclaimed-wood from a 160 years old Douglas Fir log
What I would do to "prove" this claim is look closer at the wood itself. If it's supposedly a 160-year-old log, it should be old-growth. If it's old growth Douglas fir, the growth rings should be much closer together than what you would see in new-growth wood.
For example, if you go to your local big-box lumberyard and look at a 2x4 stud, you will see very large gaps between the early- and late-wood rings, sometimes as large as 1/4"! This is because new-growth trees tend to grow very fast in their mostly treeless canopy (assuming the area they grew was clear cut), whereas old-growth trees grew very slowly in a canopy choked by other, larger trees.