Different creatures, different abilities, different rules
While all of these fall under shape-changing, each creature/ability/spell has its own set of rules.
This is a great example of specific over general with regard to the ruleset. Shape-changing is a general rule allowing a creature to change their form, while each instances of Shapechange/Polymorph has specific rules that apply.
When in doubt, do not apply general rules to a specific creature - just use the requirements and limitations stated for that creature.
Creature type vs Ability
Do note that there is a difference in the Creature tag Shapechanger and the ability Shapechanger. Creature tags are important to determine if certain spells/abilities will affect a specific creature (e.g. Polymorph doesn't work on shapechangers). The Shapechanger ability describes something that a creature can do and is dependent on their stat block with regard to how that ability works (as seen above, different creatures change shape differently.)
The Succubus
As you are looking for specific direction on the Succubus (MM, 285) for your campaign, then you need only refer to the specifics of their statblock (emphasis mine):
The fiend can use its action to polymorph into a Small or Medium humanoid, or back into its true form. Without wings, the fiend loses its flying speed. Other than its size and speed, its statistics are the same in each form. Any equipment it is wearing or carrying isn't transformed. It reverts to its true form if it dies.
Given that there is no requirement for a specific Small or Medium humanoid, it can take the form of any Small or Medium humanoid. Tricksy little fiends (note: they are not of the shapechanger creature tag, but are a fiend as stated in their stat block)!
The AC changes, to 13
In their human form, the PC weretiger with 16 DEX wears the studded leather armor, giving them 12 + 3 (= 15) AC.
They polymorph into their tiger form.
Any equipment it is wearing or carrying isn't transformed.
As you note, the armor does not transform to fit them, so they do not benefit from the human-shaped armor. It is the DM's discretion whether the armor is destroyed, damaged, or just hangs off the weretiger's new form uselessly after the transformation.
Since they are not benefiting from the armor any more, they can no longer use the AC calculation (12 + DEXMOD) provided by the armor. As Solias notes, in the Player Characters as Lycanthropes box on MM page 207, AC modifications in different forms are noted for other types of lycanthropes, but not for weretigers, so their AC is now simply the basic 10 + DEXMOD = 13.
Equipment is not a statistic
Your confusion seems to be centered around the following passage:
Its statistics, other than its size, are the same in each form.
And this is true.
If the weretiger were wearing a Ring of Fire Resistance which it took off and dropped exactly as it transformed, it would not still have resistance to fire in its tiger form, despite "resistances" being a statistic. This is because fire resistance is not a statistic of the PC, but something provided by the ring to the PC.
The same is true for the AC calculation (12 + DEXMOD) provided by the studded leather armor - it is not a statistic of the PC, but something provided by the armor to the PC.
Jeremy Crawford has tweeted to clarify that equipment is not an inherent part of monsters in the context of a similar question around the Shapechange spell - the same is true of PCs.
Best Answer
It means "don't use a different stat block".
"Statistics" is a game term defined in MM p 6-11:
So in fact the sentence you've highlighted is telling you not to use the stats of another monster like the Giant Poisonous Snake.
The mechanical difference between the forms is already embedded in the Malison stat block
The Malison specifies in its stat block which attacks and abilities are exclusive to its Yuan-ti and Snake forms. e.g. for a type 3 Malison:
So the "mechanical difference" you're looking for between the two forms is the abilities in the stat block that are exclusive to the Yuan-ti or Snake forms.