You can claim copyright on your rendition (all black versus the original color; or the text creating the logo in CSS or SVG) of the trademark in question. Whether that would hold up in court is another matter entirely.
That said, the site owner certainly holds the copyright on the site design, layout, textual content, etc. Just because it's a directory/portfolio of trademarked logos drawn by the artist doesn't mean the other content isn't covered.
That said, the site owner certainly holds the copyright on the site design, layout, textual content, etc. Just because it's a directory/portfolio of trademarked logos drawn by the artist doesn't mean the other content isn't covered.