Theme author.php transfer

You technically can, but you probably won't get the full results you're looking for. The PHP won't contain the styling, and it may contain different CSS classes than the second site's theme uses, meaning things won't look the same.

The other issue you're likely to run into is that if you put a file directly into a theme, then whenever you update that theme, your author.php will be overwritten. Anytime you make changes, unless you're using a totally custom theme already that won't be updated by anyone else, it's best practice to create a child theme.

So, to use one theme's author.php on another site, you will want to:

1. Create a child theme
2. Copy the author.php file into the child theme
3. Copy the styles that affect author.php into the child theme
4. And, most likely, do some additional CSS and/or author.php updates (such as changing div classes) so that it both keeps the look and feel from the old site, and also matches the new site.