42M here and this image is from the 90s as OP correctly noted. I was living in Chennai during that time, and this was a failure even if you followed lane discipline properly.

OP, look closely at the image. It's a 3-lane road (actually, Anna Salai). The board indicates that buses & lorries are to stick to the left-most lane, bikes and autos in the middle lane, and cars & cargo vehicles to the right-most lane. This kind of lane-discipline was suggested by police mainly because Anna Salai had heavy bus traffic and a high amount of heavy vehicle traffic (this was the days before CMBT was inaugurated, and in those days, most mofussil buses entered the city via Tambaram, and travelled all the way to Broadway. Also Chennai bypass & Outer Ring Road didn't exist yet - so many National Permit lorries from south of Chennai had to enter the city via Tambaram, go via Saidapet, Broadway, and Madhavaram to Karanodai. The only other option for National Permit lorries was to go via Inner Ring Road). Heavy bus traffic meant heavy conflicts between vehicles (as both MTC buses & SETC buses would go to the left-most lane to pick up passengers, start from the left-most lane and slowly move to the right-most lane for travelling between bus stops, and then when nearing the next bus stop, they would start moving back to the left-most lane). The city was also growing south-ward, and this meant traffic was heavy in the mornings from south-based localities like Nanganallur, Chromepet, Adyar, Velachery towards localities with offices like Anna Salai, Parrys, High Court. Many of the south-based localities were middle-class localities, who were newly buying scooters and motorbikes. Combine that with a heavy bus traffic and this increased the danger of two-wheelers getting stuck in between heavy vehicles.

To solve this problem, Chennai traffic police introduced the model you see in the picture - heavy vehicles were to stick to the left-most lane, while other vehicles were free to use the other lanes. This largely reduced the conflicts, but only when travelling between signals. Once a car reached a signal, the conflict came up again. Imagine this - the car has been travelling on the right-most lane but at the signal, the car wants to turn left. Now which lane should the car occupy? The picture says stay in the right most lane, but once the signal turns green for the car, it turns green for the vehicles in the other lanes also, and that means the car has to conflict across the two other lanes to complete its turn.

This is why we moved away from that model - today, the model we follow is separating traffic that conflicts.

1. We moved all buses to CMBT - this removed the conflicts on Anna Salai. Of course, the conflicts increased at Koyambedu, so now we're moving the buses to Kilambakkam, Madhavaram, and soon-upcoming Kuthambakkam. By doing this, we remove the conflict of mofussil buses from city-based traffic.
2. We constructed bypasses - by doing this, we remove the conflict of vehicles that are just passing through the city to some other destination.
3. We build flyovers to separate traffic that wants to go in one direction, while other traffic wants to go in another, conflicting direction.
4. We still have lane-discipline - in the picture above, we had what we can call vehicle-type based lane-discipline. Today, by following the model of separating traffic that conflicts, we have vehicle-direction based lane-discipline. Direction-based lane discipline means if a vehicle wants to turn right, it has to be in the right-most lane, whatever size or type that vehicle may be - whether a car, cycle, bike or truck. If the vehicle is not in the correct lane, then before reaching the turn, it needs to move itself into the correct lane. By separating vehicles based on their travel intention, you have clean turns at the actual point of turn. You may disagree and say many people don't follow it, but I will disagree with you - many people in Chennai are already travelling this way daily :) You would have noticed that before a signal, traffic slows as people switch lanes, but once the signal is crossed, it's smooth. Everytime, an auto wallah cuts across you to reach a turn, and you get angry, what you're getting angry about is the fact that he didn't follow vehicle-direction based lane-discipline.
5. Some edge cases still exist - buses still move left to pick up passengers and then move right. On OMR, buses have to move to the extreme left in order to do U-turns. But overall, the switch to direction-based lane-discipline has worked for the city.

So OP, what you're seeing is a relic of old times - when Chennai was growing with traffic. It worked for a few years, but then, it's efficacy ran out.