I built a new Imperial Palace today and kept my old one, so I now have two. When I look myself up on the Pedia my capital is listed as the original one. I carried on a few turns, but no change.
I think which palace actually sets the capital is effectively random.
My original capital has a supply of 3.00 from the Imperial Palace, my new Palace gives supply of 3.00, with a Space Elevator giving 0 at that location. The palace description claims a +1 bonus.
The text description is not consistent with the building script. The latter actually determines what happens.
When I check my planets I get the benefits and penalties from focus reported as my species name, not the focus name. This is quite strange for Heavy Mining, Where my population could be:
Federation Tech Subterranean Construction +12.0
Federation Tech Transdimensional Housing +4.00
I think that later on when species have bonus/penalties this should be marked by species name (maybe followed by the ability/penalty name - like technologies). The Focus bonuses and penalties should be labeled as such.
Internally, foci do not have any effects attached to them; they are just a flag on a planet, which the species' effects check for to decide when to apply a bonus or not. Arbitrary conditions can determine whether an effect applies to a planet, which may include focus, but isn't limited just to it, so there's no automatic way to assign responsibility for an effect to a particular focus. A workaround might be to have a string for effect accounting purposes with each effectsgroup, but this would be a moderate sized job to set up, and not likely to happen soon.