no it's built in. I don't know what do do. I've searched settings everywhere.
That's one of the new features of Windows 10 Mobile. You're right, it's built in.
The settings for it are kind of elusive, and can't be found through the main 'Settings' app. When you open the Messages app, from the main screen where all your conversations are listed, tap on the three dot menu, (lower right corner) and tap on 'Skype Settings'. Note you'll have 'Settings',
and 'Skype Settings' here.
'Skype Settings' is what you want. (also note: your screen may look a little different here as I've got my scaling all the way down to 175%)
Then here you can choose your default behaviors.
I don't get many Skype calls, so I don't have much experience with those, but for messaging, you can now keep all your regular SMS text messages and Skype conversations together in one convenient area. If you have a contact that has both, you can switch between SMS and Skype in the upper right corner of the particular contact's conversation window by tapping on the 'Send on' (Skype or SMS)
I'm not sure why your regular Skype app isn't working. Maybe try uninstalling and re-installing it. It should work independent of the "built in" functions I think. It's working fine on my 1520 with the Win10 Mobile build 10586.
Hope this helps!