Looks like a bug in theme engine or rather in Horizon architecture?
Here is why:
The CSS code which controls the dropdown is inside the {% stylesheet %} area in horizon/snippets/variant-main-picker.liquid at main · Shopify/horizon · GitHub
Liquid tags: stylesheet
Dox on this – Liquid tags: stylesheet
This code is rendered as part of HTML in Customizer, but on storefront its included as part of /cdn/shop/t/XXXX/compiled_assets/styles.css which is included automatically.
However, this file is not the same for different pages – looks like the theme engine selects only those liquid files which are directly involved in rendering current page.
But Quick Add modal content is fetched with JS, so this CSS is not included.
Say, if you include a “featured product” section on the same page, it will fix the problem 
Worth reporting, I guess.
Someone with Horizon in a store on a paid plan should leave a theme review with the link to this thread.
For fix – just copy contents of the mentioned {% stylesheet %} clause into, say assets/base.css or elsewhere.