Types of Battery Charging - Trickle Charging
Is trickle charging property of the charger that I plug into the wall
  socket and connect to the phone?


The phone chargers which we connect to the wall socket are AC to DC converters which provide 5 V. The trickle charging and other features are from the charger IC or section which will be put inside the mobile phones. 


  After the battery drops to some threshold (which is set by the phone or the charger?    


It is set by the phone. Think of using chargers from softener supports to different phones with different types of batteries. 

Float charging is also a property of battery management system inside the mobile phones, if they are implemented. This is not property of the charger. 




Trickle charging charges the battery with a short amount of charge ( for example C/40) regardless of the battery voltage.
Float charging charges the battery for a while and waits until a preset voltage is reached. It charges the battery to full again 
Fast charging needs to be supported by chargers as ell. The phone can ask the charger to provide fast charging over the USB cable. Charging happens at high voltage later.