Airlines have the facility legally to use a standard weight for each passenger if they wish. Most do, and for normal airliners, say 737s upward, it works OK. However, they also have the right to ask passengers to step on the scales and use actual weights. Those who fly smaller aircraft (6 - 7 seats say) and helicopters are more likely to do this. Last year I took a helicopter ride over the Grand Canyon and had to be weighed. From those weights they made up the groups and seating positions for each flight.
On larger aircraft people's weights tend to average out. Besides, although there are limits on take-off weight for the aircraft as a whole, you're talking 50 tonnes or more, so one fat passenger isn't going to make much difference. I once worked for an airline in the North of England which used a standard weight for male passengers and a lower one for females. It always made me laugh to see the tiny coal miners and their very large wives boarding!
If there is an accident where the aircraft seems to be overweight, I would suspect a gross error or maybe skulduggery, such as some heavy freight loaded but not shown on the papers. Baggage restrictions are there because small extras like 10 kgs can mount up if 400 passengers all have a bit extra (4 tonnes in that case!) and some control is necessary.
Take-off weights are a whole subject of their own. If you really want me to go into them, please ask again.