Why does setting the paper size with KOMA-Script require TikZ?

Use the option pagesize:

\documentclass[paper=a5,pagesize]{scrbook}

You may specify pagesize=pdftex or set it to dvips or auto. This option takes care of setting either pdf lengths or putting \special commands into the dvi file, see the manual.

Even without, in such cases (other classes for instance ) you don't need TikZ.

\usepackage{geometry}

takes care of the correct paper size, as well as \usepackage{hyperref}. Apparently the pdf paper size has not been correctly set (or use the default of the distribution) but geometry and hyperref fix that.

Alternatively, without any option or package this would fix it for pdfLaTeX:

\setlength{\pdfpagewidth}{\paperwidth}
\setlength{\pdfpageheight}{\paperheight}

Since KOMA-Script 3.17 option pagesize=auto is default. So the original example:

\documentclass[a5paper]{scrbook}

\usepackage{lipsum}

\begin{document}
\lipsum
\end{document}

already produces A5 sized pages.