Since your UA flight departs 2 weeks after your SQ flight, there is no chance to through check bags and you will be subject to UA's per bag limit on that flight.

Piece concept allowance does not define the maximum weight per bag. It is up to each airline to decide what they will allow. Previously most allowed 32kg (IATA recommended limit for H&S reasons), but in the past few years many airlines have adopted lower 23 or even 20kg limits per bag.