You are overcoming the static friction, which others have noted is generally greater than dynamic friction. But, the force needed at the point of contact is presumably the same when twisting versus when lifting, so why the difference in the behavior of the stand (lifting up, versus just sitting there)?
The reason is because of the direction the force is applied. Let's say a 250g force is needed to overcome static friction of the pen in the holder. If you pull upward on the pen, then if the holder is lighter than 250g, then you'll lift up the stand rather than remove the pen.
But, let's say you twist the pen hard enough to exert a 250g force on the boundary between the outside of the pen and the inside of the holder. This is applying a torque, and if the torque is strong enough the holder will twist. How strong would the torque be?
Let's say the pen outside and holder inside are 1cm in diameter, and the stand itself is 15cm across. The torque produced by your 250g at the surface of the pen is equivalent to that produced by about 17g at the edge of the holder. If the holder weighs, say, 200g, then trying to twist it with less than a tenth of that force seems likely not to budge it.
Edit: So, now you have the pen spinning; why can you lift it up without still lifting the stand? Because you've already overcome static friction; you only have to add a small vertical force to start removing the pen from the holder.
(Yes, there's a whole lot of "let's say"s in there; if you post actual weights and dimensions I'll get more concrete.)