Indeed, no current or voltage limit, this is definitely dodgy. He is using the protection circuit from a dead battery, so there is at least overvoltage protection, but this is not a charge circuit. The protection circuit is just supposed to be an extra protection in case of a failure of the charger voltage limiter. If the protection circuit itself ever fails (*) the polymer battery will directly receive 5V (minus the 0.6V from the diode) and will probably ignite.
USB chargers usually don't limit the current either and some can provide 2 amps or more. A 200mAh polymer battery won't survive long with such a treatment, but at least with such a high current the overvoltage protection will kick in pretty quickly.
Besides, how is this useful anyway? I don't remember the last time I had to change any battery in a remote but it has been years.
(*) the probability of the protection circuit failing is also higher in this case if charging with a higher current than designed for. They are usually protected against high current in the discharge path, but not in the charging path. Sending several amps through a small mosfet could destroy it and make it fail as a short, and then there would be no protection at all.