Why do Swedish postdoc ads typically have a requirement that PhD date must not be more than 5 years ago?
I don't know about Sweden, but in the US there seems to be increasing mood in Universities to end, or at least not contribute to, the "perpetual postdoc" cycle.  I don't know that there is real pressure from the funding sources, so much as a tough love from community leaders to stop taking advantage of a relatively cheap labor source by dangling prospects of academic success over them.

For example, our policy is to keep the total postdoc tenure to 5 years.  If you come in having done a 3-year postdoc, that leaves you with two more years here.  After this time limit is reached, the lab has to find means to fund you as an employee on the scientific staff if they want to keep you around.

Not all faculty love this, and some point out that by nature of some fields, postdocs tend to be longer.  For example, if you're recording from single neurons and need a large data set, this can take forever.  Despite such arguments, I haven't heard of any cases where the time limit has been waived.

The postdocs are not necessarily hung out to dry.  We have a number of programs designed to help them find satisfying employment.