I think I found the reason why it didn't seems to use my customized strings. I realized that some customization actually had been implemented on the new site but not others. The reason is that the default language pack is different on the two different versions of Moodle. In the language pack for Moodle version 3.1.1+ the Swedish translation for the string "myhome" is "Mitt hem" but on version 3.6.2 it is instead "Mitt Moodle". I had not customized that string as I was satisfied with the old default translation.