Why the Wien Oscillator output is a sine?
Here is the Wien Bridge Oscillator with a 100 Ohm trimpot slider on the LEFT side so you can INCREASE the loop gain to stabilize the output.


THus the Wien bridge resembles an LC tank circuit with high Q but at unity gain so that the output does not saturate. It accomplishes this by phase gain cancellation from Feedback RC to Shunt RC. The result resembles an LC parallel tank circuit.



If you explore the two Simulations above and add scope traces as you wish to any component or add a wire and scope that. Adjust the Pot and see how it works.  The scope trace is like a fast chart recorder but in slow motion.

This may be more instructive than my words.
HINT: Slide POT to the LEFT to reduce Rshunt and increase Gain.

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Here is another sine wave Oscillator. " The Phase SHift RC Sine OSC.  It has the same properties but is less sensitive than the Wein to gain stability and component matching.

That uses 3 Low pass Filters to obtain 60 degrees each x3 =180 degrees with negative overall feedback to achieve 0 degrees at the unity gain breakpoint and thus have a condition for oscillation.

Here is a javascript simulation of the frequency  response.  

Due to simulator's hard limiting a full swing sine can be produced. But in reality, if the loop gain is 1.001 at 180 deg phase shift, the amplitude will grow slowly and the output may clip, a condition where the OP Amp gain drops to or near zero. So soft limiting diodes are often used back to back with a high series R to not distort the sine wave yet prevent output saturation.  

Other trivia

Low Pass filters in negative feedback become high pass filters. Notch Filters with negative feedback can become BandPass filters and Negative Impedance Converters can convert capacitance current to simulate an inductor current ( within limits) using the same applied voltage.