I have also played games where we ignore 1s on Damage and Soak. But rules as written, this should not be as it.
Depending on the edition that you read, the text changes a little bit, but on most of them (I say most, because I have not read all the core books), you can find the rules of doing damage and soaking damage on step 3 of combat, called Resolution. Usually they are in the Rules and/or Drama chapter.
The rules for damage and soak are special in that they cannot botch. A Botch is considered a Fail (as in 0 success rolled).
In, for example, Mage Revised Edition (what we call 3rd edition) you have an example of soaking damage (page 238) where they roll a 1 and it does cancel one success. Examples varies across books, so not all of them will include either the actual throw (like in Mage 20th Anniversary) or not include a 1 to check this rule (like in most of Werewolf books). But Vampire Revised Edition (also 3rd), also also includes a 1 canceling out a success on a soak throw (on page 209).
Unless there are other books that cover combat (I believe there is one called WoD: Combat, that covers a lot of situations and additional rules... alas! I do not own it), or give optional rules that I don't recall (WoD does have a lot of books after all, and I haven't read them all, so... ), my best guess is that the rule "You cannot Botch" was slowly adapted by the community to mean "1s do not cancel sucess".
There are also different specific rules on different books and editions. For example, close combat success adds to damage (from revised onward) or your specialization can never allow you to reroll 10s on Damage and Soak, but none of the ones I have read (mainly Vampire/Werewolf/Mage 2nd, 3rd and 20th editions) include the rule to ignore 1s on Damage and or Soak.