I would think it's not the case for the majority of babies even. Some parents do keep the baby constantly at their side when sleeping and some are doing that weird co-sleeping thing with a raised cot right next to their own, but most parents (in modern societies where cots are used) seem to just do the usual: give the child their own place to sleep and let them develop a healthy sleeping habit right from the start.

We had our son in his own cot in his own room at night (within hearing range obviously) and in a wicker cradle during the day. We moved to an infant bed as he grew. He moved to a normal size bed a few months ago at 2½ (rather soon, but he's tall and outgrew the infant bed). He's a good sleeper.