The technology of storage batteries has changed somewhat Cyberjack, but it has become a bit more sensitive to close control, and to selecting horses-for-courses, than it was. Essentially, open lead acid batteries in the old days were "lead antimony" types with open cells. While automotive versions could not stand repeated deep discharges, they were somewhat forgiving of overcharging, so it was feasible to use them with unregulated chargers of low capacity, provided they were not left charging for too long. These days, such batteries are becoming uncommon because it was necessary to maintain them - especially, to replace lost water from the cells, and to clean off all the corrosion around them which was mostly caused by leaks of electrolyte, plus vaporised electrolyte discharged through the cell vents during charging. Nowadays, for a lawnmowing mission at a commercial level you'd use Nickel-Cadmium batteries, which not only thrive on deep discharge, they demand it. However they also demand frequent use through their full cycle of charge levels. This is good for industrial fork-lifts, but for a private mower, completely unsuitable. Lithium ion batteries may be moderately suitable but they are very expensive and will only tolerate a limited number of charge cycles - hundreds of cycles, compared with about 5 or so cycles for a car battery, but still a limited number.
Modern automotive batteries are not all that expensive and do not require (or permit) supervision of their water levels, but they will still not tolerate deep discharges or being stored with less than a full charge. Equally importantly, they must be charged with electronically controlled automatic chargers, or they will vaporise water, discharge the vapour, and run out of electrolyte since they cannot be topped up. Deep discharge storage batteries, intended for off-grid power systems, will stand a moderate number of deeper discharges than automotive batteries but they store much less energy, can deliver much less current, and are more expensive than car batteries.
Essentially, it is quite feasible to use a modern sealed Ca-Ca car battery for a lawnmower provided it is always connected to an automatic charger when it is not mowing, and it is never discharged more than say 30% of its capacity. That means you could run one of those old mowers if you wished, just by substituting a modern car battery and a modern automatic battery charger, and always plugging it into the charger when it was not actually mowing, but it would be less efficient and convenient than a modern electronically controlled electric mower. I also want to point out that it takes me about 15 minutes of engine run-time to mow all the grass I have, plus my neighbour's nature strip. This should be well under a 30% discharge for a car battery. If the hour-and-a-half of use you mentioned fully discharges the car battery, any automotive-type battery, old or new, will only last about 5 of those uses. I suspect your informant told you how long it takes to decide to mow, drag the machine out from under a pile of old newspapers and fence posts, wheel it out to the lawn, find an extension lead and run it out, remove the garden furniture and check the lawn for foreign objects, mow a bit, have a couple of beers, mow a bit more, and eventually put everything back as it was except for the grass being shorter.
As a piece of useless information there was a type of battery in the old days that would stand quite a few deep discharges - more than a modern deep cycle battery. It was known as an "ironclad" because of its relative indestructability, and was used in submarines. However they were stupendously large and heavy for their capacity, and AFAIK were never used significantly for civilian purposes.
