Do plants have distinctive DNA genomes from each other like humans do?
Short answer: Yes!

Now for the longer answer. Yes, they do have distinct genes in general, but what you may see as different plants can also sometimes be clones. For example Quaking Aspen Trees which commonly grow in clonal colonies.

If you had a massive database of genetic material from plants you would be able to trace it. You may be able to, by doing sampling, work out where a sample was located geographically.