Got behavior, when my pre-push config was declared as:

// package.json
{
  ...,
  "scripts": {
    "prepush": "npm test"
  },
  "husky": {
    "hooks": {
      "pre-push": "npm run prepush"
    }
  }
}

After investigation understood, that this was my mistake:
Husky run both husky.hooks.pre-push AND scripts.prepush.

But it was annoying that tests were initiated twice and no warnings/errors were shown about duplicating config declaration.