Write sed command to input file

To add a single key-value pair to the scripts object:

jq --arg key 'compile' --arg value 'browserify js/main.js > ./dist/bundle.js -t babelify' \
    '.scripts += { ($key): $value }' file

Adding two keys and values, just pipe the output through exactly the same jq expression again, but with other values for the internal $key and $value variables:

jq --arg key 'compile' --arg value 'browserify js/main.js > ./dist/bundle.js -t babelify' \
    '.scripts += { ($key): $value }' file |
jq --arg key 'watch' --arg value 'watchify js/*.js -o ./dist/bundle.js -d' \
    '.scripts += { ($key): $value }'

Given the document in the question, this would produce

{
  "name": "torrent-search-api",
  "version": "1.0.0",
  "description": "",
  "main": "index.js",
  "scripts": {
    "test": "echo \"Error: no test specified\" && exit 1",
    "compile": "browserify js/main.js > ./dist/bundle.js -t babelify",
    "watch": "watchify js/*.js -o ./dist/bundle.js -d"
  },
  "author": "",
  "license": "ISC",
  "dependencies": {
    "torrent-search-api": "^2.1.4"
  },
  "devDependencies": {
    "@babel/core": "^7.16.7",
    "@babel/preset-env": "^7.16.8",
    "babelify": "^10.0.0",
    "browserify": "^17.0.0"
  }
}

You would redirect the output of the second jq command to a new file and then move that file to the old name, like one usually does.

You could obviously do both operations in one go if you know about $ARGS.named, an object containing the named variables on the command line:

jq --arg compile 'browserify js/main.js > ./dist/bundle.js -t babelify' \
   --arg watch   'watchify js/*.js -o ./dist/bundle.js -d' \
   '.scripts += $ARGS.named' file

The result is the same as with the pipeline using the two more naive commands, and depending on your script and other circumstances, you would either want to do it that way, or like show just above.