Link relevance is not only important, but it is also absolutely essential to ranking. However, the way you framed the question is flawed and shows a fundamental lack of understanding of how search engines work.
Search engines index and rank individual web pages, not websites. Therefore it doesn’t matter what your website theme is, it has absolutely nothing to do with ranking. However, page relevance is what ranking is all about. So, while website themes have no direct relevance to SEO, page topics are of paramount importance.
To understand the role link relevance plays in SEO it is important to recognize how Google determines link relevance, not your own definition, but how Google defines relevancy. Google uses proximity as a weighting factor in determining relevancy factors, so the closer relevant keywords are linked to your page the higher your relevancy score will be influenced.
Keywords in or near the anchor text of links influence the relevancy of your link. Keyword proximity places a significant role. The closer the keyword is to your link, the stronger the relevancy signal. By placing links on relevant pages with your targeted keywords in or near the anchor text you are creating relevant links. There are other factors that affect signal strength, including PR and trust.