In percentage, how much is the human genome (DNA) similar to the mouse genome?
This question cannot be answered as simply as you put it, but it's not too much to elaborate on.

The order of the base pairs will be drastically different, but the same proteins and amino acids will be coded for in genes, just at different points along the DNA. For example, you may find the same sequence to code for a protein in a mouse as in a human, but they will not be found in the same place along the DNA code (although they will still be read and eventually produce the same protein).

http://www.nature.com/nature/journal/v420/n6915/full/nature01262.html
Additionally, this article shows that, if we take your question literally, "at the nucleotide level, approximately 40% of the human genome can be aligned to the mouse genome", that is, if human and mouse DNA was lined up side-by-side, and a percentage of identical match-ups was calculated.

However, taking a 'looser' approach, it is worth noting that about 5 percent of both of genomes consist of protein-coding genes (https://www.genome.gov/10001345), and the remaining 90%+ is non-coding. If we only count the protein-coding genes, the average similarity of genes between humans and mice in only protein-coding genes is 85%. The link above says that some are 99%, and some are 60% - there is a clear similarity here between humans and mice, albeit not 100% identical.

If you wish to know the explanation, simply it is because mice and humans both have a common ancestor (the last link also explains this) about 80 million years ago. Due to these similarities, and the striking similarities, scientists simulate genetic changes in mice that can occur to humans, and have used computers to analyse the similarities in mouse genome.

I hope this is understandable, if you need any clarification on terms, please ask :)