Okay I found a way to add your own logo. Probably pretty hacky but it works.
Open one of these files depending on the theme you're using:
/node_modules/@elastic/eui/dist/eui_theme_light.css
/node_modules/@elastic/eui/dist/eui_theme_dark.css
Find .euiHeaderLogo__Icon and change opacity to 0. This will hide the original logo.
Next look for euiHeaderLogo, should be right above .euiHeaderLogo__Icon. Add some css such as background-image and background-repeat to add your own logo.
Now look for .euiHeaderLogo:focus and delete the line background: #232635; } or your logo will disappear when you click it.