It is not possible to simply customize updown.io status pages design or logo for the moment. You can find a couple suggestions here if you want to add your vote:
https://updown.io/features/custom-logo-for-public-status-pages
https://updown.io/features/custom-css-for-public-status-pagess
It is, however, possible to remove the updown.io logo using the White label option in your settings:

Also the description field of your multi-site status page can include markdown (including images), which can help you make it look like yours (e.g. showing your logo/brand/client logo).
Finally, it is always possible to use your read-only API key to build truly custom status page or dashboard in javascript if you need something different. Our API accepts CORS requests so you can built it in standalone javascript.
Here are some example custom status page made by our users that you could start from:
https://github.com/nya1/updown-status-page
https://github.com/Hexstream/status.abc.hexstream.xyz
https://github.com/SocialGouv/dashlord
https://github.com/MTES-MCT/updownio-action