No, you are neither legally allowed in most countries to share a lane with a car (some exceptions exist) nor do motorcycles in general tolerate cars well coming into their lane.
I drive myself motorcycle since almost 20 years. Most EU countries do not allow a motorcycle sharing a lane with a car (one exception e.g. Netherlands), what you observe are mostly impatient motorcycle drivers who take the risk weaving between two lanes to get faster forward. In fact this strategy is very common if a green light has only a short time and a queue is building...which gets longer and longer. This can also be observed on autobahns when the traffic is jammed and the cars are motionless or driving extremely slow.
strictly in many countries illegal, this kind of driving (if careful) is tolerated by most car drivers (and the police often also look the other way) because you allow the car behind to skip your position. Also you are always the vulnerable person on the motorcycle; if something happens, you are likely to be (heavily) injured or killed. For this reason, it is always the motorcycle, not the car who is maneuvering in a lane.
Now motorcycles are often driving in convoy; during this the motorcycles are driving with two in a lane in a zig-zag pattern which gives optimal distance and close into two columns when the convoy comes to a stop.
Don't move close to a motorcycle with a car, this is a surefire way to aggravate the driver. You simply have no crush zone as motorcycle driver and the mass advantage means that even a slow collision can break or squash legs.