The other question is, what happens in general when Google finds a completely new site on my old domain, with similar or close to the same content?
If you are doing a design/styling transition and content is going to remain the same on the existing domain:
When the design/styling is CSS, then I prefer to do a few pages and see how the search engines absorb it. In other words, content remains mostly the same, stylesheets in the page header point to stylesheet2.css instead of stylesheet1.css.
When the transition is from static html to a cms such as WordPress, it gets more complicated. What I have done in the past is set up WordPress in parallel to the existing site html. While it sounds complicated, WordPress probably isn't going to be using any of your existing folder or page names. While I shouldn't have to say it, backup before starting this kind of thing.
If you have an existing example.com/page1.html, in the WordPress cms, the same content can be uploaded as a page named example.com/subject-item.html. Or whatever you find suitable.
Then, using htaccess, permanent redirect example.com/page1.html to example.com/subject-item.html
You will see some drops in ranking while the search engines digest the changes, but if every page on your site is properly redirected, there won't be any long term impacts.