Is this object an asteroid or comet, and how can it produce so many tails?
A paper titled "Monte Carlo Models of the dust environment of a sample of comets from the Oort Cloud to the outer Main Asteroid Belt" (PDF also here) (2014) studied the behaviour of P/2013 P5 using models and concluded that


  
  The object has been subjected to an intermittent dust mass loss, most likely associated with a rotational disruption. This is confirmed
  from the analysis of both HST and GTC images. The total dust mass
  released was of the order of $10^7\ kg$, for particle density of
  $1000\ kg\ m^{−3}$ and geometric albedo $p_v = 0.04$.
  The model of rotational disruption, based on simulations of an object that loses mass from its equatorial region, and whose
  rotational axis is perpendicular to its orbit plane, reproduces to the
  last detail the observed complex brightness pattern at four different
  epochs of HST and GTC observations. For obliquities different from 0°
  or 180° , the fits get much worse. On the other hand, an isotropic
  ejection model does not fit the HST data because it produces much more
  diffuse tails than observed.
  The ejection velocities are very low, of the order of $0.02–0.05\ m s^{−1}$. This places the limit on the size of the object to be in the
  range $30–134\ m$ for assumed densities of $3000–1000\ kg m^{−3}$.
  


This seems to confirm what the original question cited as a possible explanation, i. e. 


  The tails are suspected to be streams of material ejected by the
  asteroid as a result of a rubble pile asteroid spinning fast enough to
  remove material from it.


The asteroid is loosely bound (a "rubble pile") and spinning fast, so everytime some of the boulders which form it shift a bit, some material may escape into space and form a small tail.