The Skatteverket ID is indeed not from the Swedish police, and what Skatteverket (Swedish IRS) require to issue ID is different from the Swedish police. The ID from Skatteverket seems to be primarily used to get non-Swedish citizens some Swedish ID (since even the Swedish police refuse to accept valid foreign passports as ID under a variety of circumstances).
The Swedish intra-Schengen border checks were until now being approved on a 14 day basis and extended every 14 days; but now they will be approved for a month at a time. So this is now approved to continue at least until February 8th and will then be subject to approval for another month and another month and another month and so on.
So much for the EU/EC standing up for intra-Schengen mobility rights.
As the number of new refugees filing in Germany daily is still in the 3000s per day, intra-Schengen mobility is going to remain in a position to be compromised at this level or more unless and until one or two conditions are met in/near Turkey. And the conditions driving up the demand to leave SW Asia and even parts of Central/South Asia are getting worse and worse due to international power plays. Good luck, Schengen; you'll need it. And good luck, refugees; you'll need it too.
While the latest Swedish and Danish checks are helping out Norway, Finland and maybe even Denmark when it comes to reducing the number of incoming refugees for now, the larger dynamic that creates refugees in SW/C/S Asia isn't changing well enough to alter the demand to leave the region. In other words, these intra-Schengen checks are going to remain in this region for quite some time unless and until they shift to other parts of the Schengen Zone and conditions on the ground elsewhere end up being more peaceful than is currently the case.