You can incorporate MIT/BSD code in a proprietary project, but that imported code itself remains BSD/MIT licensed. For many projects, this is a technicality, but no, you can't claim copyright on MIT/BSD code that isn't yours.
But also nothing preventing you from doing it.
MIT can be relicensed, for sure.
The MIT license is not very long, and it contains this sentence:
> The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
Which clearly state that you can't remove that license to put your own.
Probably the sublicense wording that many techies confuse with relicensing.
Also sloppy wording in discussions like HN where "relicense" is sometimes used as a shortcut for "combined with a compatibly-licensed work in a larger work that can be used under the whole-work license".
That is not the case with other licenses.
> Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
> The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
Note the "deal in the Software without restriction" and "sublicense" permissions and the virality of the terms that require the permission notice must be included with all copies.
One problem with this however is if someone has removed the license from the software like what has happened here, then that software really has no license because the license doesn't specifically state that it applies to derivatives. The Apache 2.0 license is much clearer on that subject of derivative works.