Online file hosting. I click "Upload File". I select correct file. I click "Open". File gets uploaded. I log out.

What else does hosting get to know?
(besides name, modification timestamp, creation timestamp, content, extension and length of upload file etc.)

Assuming running Linux Ubuntu/Arch/Whatever, on Mozilla Firefox or any other open-source browser with no obvious exploited vulnerabilities.

That's a vague question. But the file was sent as it was uploaded so all that's there is there. That is, the old meta data in files have been a curse to some. Example:

Your electronic files may include "metadata"—previous drafts, revisions, and even comments meant for exclusively internal use. The metadata associated with your documents may contain confidential or privileged information. It may also include information damaging to your position in the matter at hand.

In addition to basic information such as your name, initials, firm name, names of previous authors, and where the file was saved, the metadata may include "redline" revisions in the document, hidden text, and comments. This article tells you how you can protect yourself from metadata issues in Microsoft Word and WordPerfect.

If the file isn't encrypted (you did not specify) then they have access to the entire contents. I the file is encrypted but the file name is not then perhaps inferences can be made from the file name.