The convention is to have the testsuite to write to a path given by the LOG_DIR environment variable (CI / Quibble set it). CI then just capture whatever is in that path and publish it as a build artifact.

An example from the webdriver.io configuration file:

const logPath = process.env.LOG_DIR || path.join( __dirname, '/log' );
...
reporters: [
  [
    'junit': { outputDir: logpath }
  ]
]

@Jpita the easiest way to save cypress screenshots and videos is to save them to a location that Jenkins already archives. I'll take a look.

this is easily done by adding:
config.screenshotsFolder = process.env.LOG_DIR || path.join( __dirname, '../log' );
and
config.videosFolder = process.env.LOG_DIR || path.join( __dirname, '../log' );

to the cypress/plugins/index.js file.
Example here and here.

@Jpita sorry, it's been a while since you've reported this. I think this could be resolved like this:

tests/cypress/plugins/index.js

//...
config.screenshotsFolder = path.join( logPath, 'screenshots' );
//...

See 601343 and sample.spec.js.mp4.