Do scenes store controller assignments?

No, controller assignments are stored in the Performance. Each Part has its own set of controller assignments. Referred to as CONTROL SETS. They include a Source (controller), and a Destination (parameter)... and a set of parameters that describe the application of this control (shape, depth, direction). There are 16 Control Sets per Part. And 16 more Source/Destination setups at the Common (overall) Performance level.

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Mastering MONTAGE/MODX: SCENES

A Motion Control Synthesis Engine way to look at controller assignments would be — based on the fact that you have simultaneous direct control over as many as 8 Parts with your array of standard controllers, and because what a controller is doing can differ from Part to Part, and because you can change, via Scenes, which of those 8 Parts you are addressing at any time (KBD CTRL active/inactive status is stored per Scene)... so, yes, you can, if you approach it from this vantage point.

What are you ultimately trying to accomplish?

Unless you consider Superknob to be a controller. Other controllers cannot be stored, but superknob can be.

Superknob can control assignable knobs (and Yamaha documentation shows assignable knobs as controllers). So those can be set by scene indirectly by setting the position of superknob in a scene.

Neither superknob nor assignable knobs may be "controllers". But I'm not sure what the OP's definition is.

Yamaha's data list shows:

BREATH CONTROLLER 2
FOOT CONTROLLER 1 11
FOOT CONTROLLER 2 Super Knob
ASSIGNABLE KNOB 1 17
ASSIGNABLE KNOB 2 18
ASSIGNABLE KNOB 3 19
ASSIGNABLE KNOB 4 20
ASSIGNABLE KNOB 5 21
ASSIGNABLE KNOB 6 22
ASSIGNABLE KNOB 7 23
ASSIGNABLE KNOB 8 24
RIBBON CONTROLLER 16
ASSIGNABLE FUNCTION 1 86
ASSIGNABLE FUNCTION 2 87
MOTION SEQ HOLD 88
MOTION SEQ TRIGGER 89
FOOT SWITCH Arp SW

As assignable "controllers". So Assignable Knobs do make the cut of "controller" in a general sense.

The "reset all controllers" function resets:

PITCH BEND CHANGE 0 (center)
CHANNEL AFTER TOUCH 0 (minimum)
MODULATION 0 (minimum)
EXPRESSION 127 (maximum)
BREATH CONTROLLER 127 (maximum)
FOOT CONTROLLER 1 127 (maximum)
FOOT CONTROLLER 2 127 (maximum)
FOOT SWITCH 0 (off)
RIBBON CONTROLLER 0 (center)
ASSIGNABLE FUNCTION 1 0 (off)
ASSIGNABLE FUNCTION 2 0 (off)
MOTION SEQ HOLD 0 (off)
SUSTAIN SWITCH 0 (off)
SOSTENUTO SWITCH 0 (off)
MOTION SEQ LANE 1 0 (minimum when Polarity is set to Unipolar and center when Polarity is set to Bipolar)
MOTION SEQ LANE 2 0 (minimum when Polarity is set to Unipolar and center when Polarity is set to Bipolar)
MOTION SEQ LANE 3 0 (minimum when Polarity is set to Unipolar and center when Polarity is set to Bipolar)
MOTION SEQ LANE 4 0 (minimum when Polarity is set to Unipolar and center when Polarity is set to Bipolar)

.. so there may be more things to add to the list of "controller" from above.

Although super knob "looks" like a controller, it doesn't seem to be defined as one anywhere (... that I just checked). I haven't done an exhaustive search.

... also, there's "controller box source" as part of the motion control destination modulation matrix. The controller box sources show "envelope follower" as one of these "controllers". And you can set, in the scene mixer settings, the levels. Therefore, you can manipulate envelope followers indirectly by storing levels (in terms of offset/scaling) that those envelope followers are following (PARTs 1-8 or master).