I´ve create a custom search. The results shown a Picture, Post Title with Link and some custom field information. I´d like to linked the picture and in addition to include a separate Button as a link. The target of both links should be the same as the Link-Target of "Post Title with Link".
To create a button which matches the theme style you need to inspect the markup generated for a button by your theme and then reproduce the same markup (including the class names) where you generate the button.
In your browser right-click the button and Inspect it.
I don't know what theme you are using, but doing that in Divi, for example, shows the following markup for a button: