I only know the situation in Singapore, as I have used both taxis and Grab for getting around with a baby.
The basic rule is: any person with a height of below 1.35 metres must be seated in a child restraint or a booster seat. There is however an exemption for taxis, where children are allowed on the rear seat without a child restraint.
This exemption does not apply to private hire vehicles such as Grab though. In that case you have the option of bringing along your own car seat, or use the special Grab Family option, that comes with two sizes of car seat. There's an extra charge of $5 or $2 depending on the age of the child. I've never actually used this option, so I'm not sure how well it works in practice.