Generally, subdomains are associated with the parent domain as an additional part to your main domain name. Subdomains are created to organize and navigate to different sections of your website. Like a department store.
store.example.com <-- Ecommerce part of my business site
en.example.com <-- English version of my business site
cats.myblog.com <-- Spin off of my personal blog dedicated to Cats
Subdomains can be about completely different topics from a main site and not affect the way that Google ranks one or the other. The association is the entity tho, I would not put sites I don't own on the same domain as my main site.
In your case, yes, I think Google will associate those sites with your app site, which could affect your SEO in a variety of ways.
I would recommend not giving your users addresses that are on the same domain as your application's site because like you said, you have no control over what they host per the nature of your service.
There is not really a good (or reliable) way to tell search engines that those sites should not be associated with yours. Also, the user subdomains are affected in the same sort of way.
Google might figure it out eventually, but at what potential cost? It probably is helping that - they're on separate servers from yours tho.
Some Other Challenges:
- How/when will you know if this is affecting you?
- Links to your users' sites will contribute to your domain's backlink profile - the link equity earned by a subdomain will not impact the root domain in the short term but in the long term it will.
- Subdomains are often an SEOs choice method (over subfolder) of internationalization because they inherit equity from their parent directory and are useful for providing backlinks (seen as external, not internal) to the main root website.
- However, in your situation, this would make your backlink profile a disorderly mess.