This one seems to be difficult to find any sources on, in part because it's inherently impossible to prove a negative. While there may be some sources I'm unaware of, here's a general overview of submarine tactics during WWII.

Submarines were primarily used as either a first-strike weapon, as in the Battle of the Atlantic against Allied convoys or the Pacific campaigns against the Japanese. Submarines were not particularly sophisticated, and unlike later designs, they were noisy and slow. The use of diesel engines meant that the submarines were very loud, and not particularly well adapted to counter-submarine warfare, which was left to surface forces. The Royal Navy used destroyers extensively in anti-submarine operations, and as can be determined from Lothar-Günther Buchheim's Das Boot, written during his cruise on U-96, the German submariners were used primarily to attack convoys and retreat to avoid destroyer countering. The depth-charge attacks so often shown in movies like Das Boot and U-571 really happened, and destroyers also rammed submarines wherever possible, in both oceans - the HMS Dianthus sank U-379 by ramming, as you can see in this report and in the 1st World War, the RMS Olympic sank SU-103 by ramming.

In short - no, submarines did not typically engage each other. They were vastly more concerned with attack from either fast surface forces or from the air. You're correct in that the technology of later Cold-War standoffs did not exist during WWII, so the submarines could not effectively engage each other. That's not to say they didn't, but they did not specifically send out submarines for the express purpose of counter-submarine warfare; submarines were used to strike slow surface fleets.