Is Bioshift affected by Doubling Season?
Yes it does. See the rulings on Bioshift:


  To move a counter from one creature to another, the counter is removed from the first creature and placed on the second. Any abilities that care about a counter being removed or placed on a creature will apply.


The ruling stems from the following rule:


  121.5. If an effect says to “move” a counter, it means to take that counter from the object it’s currently on and put it onto a second object. If the first and second objects are the same object, nothing happens. If the first object has no counters, nothing happens; the second object doesn’t get a counter put on it. If the second object (or any possible second objects) is no longer in the correct zone when the effect would move the counter, nothing happens; a counter isn’t removed from the first object.