I can't find any information on how Malmö and Copenhagen were connected (and who the operator was) before the Öresund Bridge was inaugurated.
How were they connected? Generally they were not.
There were car ferries, but they were slowish and mostly used as part of long-distance journeys rather than just between Copenhagen and Malmö. The last several decades they didn't connect anywhere near the city centers either. Either you would drive to Dragør (outside Copenhagen) for a ferry to Limhamn, or you could take a ferry from Tuborg (within Copenhagen) to Landskrona (quite a distance from Malmö).
One company, Flyvebådene, operated fast passenger ferries (not taking vehicles) between the city centers. These had some commuter use but nothing that figured appreciably in the average Copenhagener's mental map. I think people on the Swedish side were generally more aware of Copenhagen than vice versa, though.
As Tor-Einar Jarnbjo points out, most road traffic between Zealand and Sweden used the (still existing) Helsingør-Helsingborg ferry crossing about 40 km north of Copenhagen.
Just an addition to Henning's answer, but too long for just a comment:
It is perhaps worth noticing that before the Øresund bridge was built, all rail and most road traffic from Copenhagen and Zealand to Sweden (in general) used the ferry connection between Helsingør and Helsingborg, about 50km north of Copenhagen. This traffic did not go through Malmö at all.
If you are going by car from Copenhagen to anywhere in Sweden north of Malmö (which means almost anywhere in Sweden), it can actually still make sense to use the old ferry connection, which is still fully operable. Since the bridge enters Sweden a bit south of Malmö and the motorway continues quite a bit south and then east of Malmö before turning north, the road distance from Copenhagen to e.g. Gothenburg is about 40km longer via the bridge and Malmö compared to the old route, so if you plan about 30 minutes for the ferry crossing, you can save yourself a 40km drive. The ferry ticket and the bridge toll is about the same.