What it means
In British English, lift pulls double duty. It’s the thing that takes you up the building, and it’s also a ride in someone’s car. Context does the heavy lifting. If you’re in a block of flats or an office, you take the lift. If your mate drops you at the station, that’s a lift too.
Usage examples
"You taking the lift or the stairs? And mate, could you give us a lift to the station? It’s chucking it down and the bus is nowhere to be seen."
"Take the lift to the fourth floor, the stairs are blocked."
"Any chance of a lift to the station? It's chucking it down."
"Missed the last bus, so my sister gave me a lift home."
"The lift's out again, so if you're going to the sixth floor, good luck to your knees."
Where it comes from
Both meanings grow out of the old verb lift, meaning to raise or move something upwards. The building sense is the British word for elevator, first tied to machines that lift people between floors. The car sense came later as an informal favour, lifting someone along on your route.