This app was mentioned in 2 comments, with an average of 2.00 upvotes
I looked into this awhile ago and decided it wasn't worth it. https://play.google.com/store/apps/details?id=mobi.infolife.smsbackup
You have to export and tell the app who each conversation had in it.
For the future, you can use this to backup SMS. I don't believe it does MMS at all, but you can save those photos easily. I've found this is the easiest and fastest way to back them up (it creates a file you can copy off of your device and then put it back on after a wipe).
https://play.google.com/store/apps/details?id=mobi.infolife.smsbackup
I'm doing a test run of it on Marshmallow right now. The Play Store says it works on KitKat, but I've definitely used it on Lollipop, and I'm doing a test on Marshmallow right now. It looks like it works.