Three elements shape how recognisable your listing is, and each is influenced differently.
Site name
Google shows a site name rather than a domain for the homepage and can use it elsewhere. Influence it with WebSite structured data carrying a name, a consistent <title> on the homepage, and Open Graph og:site_name. Be consistent across all three.
Favicon
Shown next to results on mobile. Requirements: a square multiple of 48px, at a stable URL, referenced with a rel="icon" link, crawlable, and visually representative of the brand.
Sitelinks
The extra links shown under some results are chosen algorithmically. You cannot specify them. Clear site structure, descriptive internal link text and a logical hierarchy make good ones more likely; there is no markup for it.
Alien Road recommends: use one favicon file at one stable URL and stop changing it. Google caches favicons and re-crawls them infrequently; sites that rotate the file through a build hash often show a blank or stale icon for months.