Say I have a multiclass warblade/swordsage. I want to go into Master of the Nine.
What are my initiator levels for each class? How do I prepare maneuvers gained through the PrC? Under what classes do/can they go?
From Tome of Battle page 96,
If you have levels in two or more martial adept standard classes (for example, you are a multiclass swordsage/warblade), you must decide to which of your existing martial adept classes the new maneuvers known or maneuvers readied apply. [...]
Maneuvers Known: When you gain additional maneuvers known, these simply add to the maneuvers known of one martial adept standard class you already possess. [...]
Maneuvers Readied: [...] If you have more than one martial maneuver progression, you must choose which progression the additional readied maneuver slot applies to. [...]
Recovery: [...] If you have levels in more than one martial adept class, you choose which recovery method you will use based on which adept class the new maneuver you are learning applies to. Once this choice is made, it cannot be changed.
So for each maneuver known or maneuver readied, you have to pick one class. It will be recovered based on the recovery method used by the class it gets assigned to. You can choose differently for each maneuver known or readied you gain.
For initiator level, things are slightly murkier. The multiclass characters section on page 39 makes a brief reference to them:
Prestige classes work a little differently. In most cases, you add the full prestige class level to your martial adept level to determine your initiator level. See the prestige class descriptions in Chapter 5 for details.
Unfortunately, “most cases” is ridiculously vague wording, and despite having just spent two paragraphs describing how different initiating base classes have separate initiator levels, this refers to a single initiator level rather than explicitly stating anything about separate levels. Actually, arguably, the most useful thing in this paragraph is that it explicitly describes the situation as being different from the prior paragraphs about base classes.
Really, though, page 39 just relies on its reference to “Chapter 5,” which starts on the aforementioned page 96, to actually supply real details. Unfortunately, there is no mention of initiator level in the general prestige class rules in Chapter 5. You have to look to individual prestige class descriptions, such as for master of nine:
You add your full master of nine levels to your initiator level to determine your total initiator level and your highest-level maneuvers known.
And that is rather light on details: again, no addressing of exactly how this works for multiclass initiators. So despite page 39’s directive to look here, we still do not really have any explicit statement about how this works.
However, while I would have liked things explicitly spelled out, we can still determine what these rules add up to if we consider the process here: when do you determine your initiator level? When learning new maneuvers, or when a maneuver bases something on it. So say you choose to add a new maneuver known as a warblade maneuver, what initiator level do you have? The rule above says you add your full master of nine level to your warblade initiator level. But what if you add a maneuver known as a swordsage maneuver? Same deal: when you go to determine your swordsage initiator level, you add your full master of nine level.
In short, while you have to pick and choose which base class a maneuver known or maneuver readied goes to, you don’t have to do any such thing for initiator level: the prestige class adds itself, in full, to any and all initiator levels that you have. No rule mentions picking one, and no rule describes ever adding only half of your prestige class levels to any initiator level. Furthermore, page 39 did helpfully tell us that we should expect prestige classes to handle initiator level differently from base classes, which this is.
So a 5th-level warblade/2nd-level swordsage/5th-level master of nine would have warblade initiator level 11th (5 from warblade + 1 from swordsage + 5 from master of nine), and swordsage initiator level 9th (2 from warblade + 2 from swordsage + 5 from master of nine).
You didn’t ask, but to finish this out, back on page 96 we also have
Stances: If a class grants new stances known, you can select stances from the disciplines available based on the prestige class.
And that’s it. This is because stances do not have any reason to be associated with a particular class: they are not expended or recovered, they don’t have to be readied, you just know them. Some of them do scale based on initiator level; presumably you are free to use whichever initiator level you like for them when you use it.