Let's go step by step.
Does the paladin mount advance when the paladin is multiclassing?
No, there's no other class that keep evolving the paladin special mount. The multiclass of Ranger and Paladin have access to the feat Devoted Tracker (CAdv 107) though that while doesn't keep evolving it allows you to choose your Special Mount as an Animal Companion and get both bonuses simultaneously.
Does the paladin mount advance when taking levels in a prestige class?
The standard answer is no, but there are exceptions. One of those is the Cavalier PrC (CW 19) that explicitly says otherwise: "Special Mount: A cavalier's class levels stack with any paladin levels the character might have for determining the characteristics of a paladin's mount."
Does the paladin mount advance when taking levels in the Fist of Raziel class?
No. The Fist of Raziel specifically says it doesn't advance the special mount in the section about its Spells per Day as Taormina quoted here.
If the answer to the questions above is no, would it be irrational to house-rule that prestige classes that work well with the paladin class could also count on the table for paladin mount advancement?
Not really if the class in question is focused in the mount aspect like the Cavalier mentioned before. Notice that PrCs usually specialize in certain abilities/acquire new abilities and discard others in order to follow its concept and keep as balanced as possible (I'm looking at you non-druid PrCs with full casting and several abilities. Shame on you!).
Then a class that "work well" with Paladin that has a focus on mounts could advance it. However, if it also advances Smite Evil, Turning Undead and spells per day it would be "working well" too much like an upgraded Paladin 2.0 instead of a path for the Paladin. So I would rule out most Paladin PrCs since they usually focus on Smite Evil or another core Paladin aspect. Although a PrC that gives at least one ability for mounted use like a the Mounted Combat feat (not as an option of bonus feat, but as fixed feature), different special mounts or another feature exclusive for mounted combat would probably be OK if they don't already do it.