In the videogame series, Silent Hunter, which simulates being a submarine captain in WW2, after a patrol you are told how much tonnage you sank. I always thought that was just a reward mechanism to keep you playing. But in this List of U-boat Aces, it lists the tonnages each ace sank. It does note the caveat that this may be inaccurate due to multiple U-boats claiming the same kills, but it doesn't imply any inaccuracy in the size of the ships.

So how do we know how much tonnage they sank? Are these numbers gathered post-war or is there some method for measuring a ship's tonnage through a periscope?

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    No, you just look them up in Lloyd's Register of ships – sempaiscuba Feb 13 '20 at 19:49
  • @sempaiscuba and the Germans would've had access to that? Despite it being a British publication? – Ryan_L Feb 14 '20 at 4:51
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    Yes. They would have copies up to 1939 when the war began. Even then, the German government could get up-to-date copies in the same way they got copies of British newspapers - via neutral countries. – sempaiscuba Feb 14 '20 at 9:03
  • What research have you done? Like most SE sites, we prefer questions that show what has been done to solve the question in advance; it prevents us from searching the same sources, and frequently reveals assumptions that complicate the question. – MCW Feb 15 '20 at 14:36

Any submarine would try to identify the target before firing. While mistakes are possible, both regarding the proper identification and the listed tonnage, reports would list what they thought they were shooting at.

For warships, the best-known reference is Jane's Fighting Ships. For instance, the 1978 edition lists 7 ships with the hull number "16", one "D 16", one "DE 16", and three "F 16" in various navies. So a ship spotting an "F 16" would have to check if it was the Diomede, the Umar Farooq, or the Oland.

Information got somewhat harder to get during wartime, as discussed in the comments to the OP, but navies produced manuals like FM 30-58, the US guide to Japanese warships.

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