This is an interesting question.
Formally, yes, when you tear the paper, there is sensitive to initial conditions, and that explains why it is very unlikely that two pieces of paper are teared exactly in the same shape on every tear.
However, the fact that there is chaos does not imply that something can not be described macroscopically. There is nothing forbidding that macroscopically something is very likely to happen (i.e. the piece always end up in two pieces).
Let us consider a more familiar example, friction. A ball thrown to the air will suffer friction. Microscopically, particles are hitting the ball, making it to decrease its velocity. However, the trajectory of the ball can still be well approximated by Newton's law + stokes drag. Why is that?
Chaos does imply that, microscopically, the system is not integrable (and thus no long term deterministic predictions of the state are possible), but it does not imply that it can not be described statistically(1), and that its most likely outcome is so probable that the most likely outcome behaves as if it was deterministic (e.g. stoke's law).
More formally, what happens in my ball example is that the it is statistically extremely likely that the ball will follow the path described by Newton's law + stokes drag, even though the specific positions of some of the particles involved (air particles) will still be very different, due to chaos.
Likewise, in the example of yours, what happens is that it is statistically very likely that the pieces will tear in two pieces such that you end up saying "I always end up with two half pieces".
In summary, chaos implies that trajectories diverge exponentially in time and that the (microscopic) state of the system is unpredictable, but it does not imply that the evolution of an ensemble of unpredictable trajectories is unpredictable. In particular, it can happen that it is extremely likely that particles will almost always end in two separated pieces.
(1) Statistically here means over an ensemble of initial conditions. In your case, the ensemble of the different microscopic ways you decide to tear the paper.
Going a step forward in your example, you could decide to sharply fold the paper prior to tear it, for the paper to be teared along that specific fold line. What you are effectively doing is to increase the likelihood that the paper will be teared over that specific fold line.
A crucial pillar of statistical physics is that "increase likelihood to be in a specific configuration" costs you energy (e.g. to fold the paper). In physics jargon, it costs energy to decrease the entropy.