Open source license does not mean giving up intellectual property rights. You can only license the work if you own the rights to the work.
MIT License starts with copyright notice:
Copyright (c) <year> <copyright holders>
Permission is hereby granted, free of charge, to any person
obtaining a copy of this software and associated
documentation files (the "Software"), to deal ....
It's standard to use both “All rights reserved” along with explicit permissions to be explicit that all rights under copyright not explicitly granted are reserved. It is, in intent, a disclaimer of additional implied permissions, though IIRC the “All rights reserved” in practice has no legal effect.
MIT is how the work can be used by others. This is basically saying "hey, I claim all the intellectual property rights granted to me by law. However, if you use this software strictly under these terms (MIT), I will not consider it to be infringing upon my rights."
No, it's generally "Copyright <name>" to explain where the license is coming from, but a copyright license is the exact opposite of "all rights reserved".
MIT License starts with copyright notice: