Okay, my mental model of identical? is not accurate. It calls clojure.lang.Util/identical which has the following implementation:
static public boolean identical(Object k1, Object k2){
    return k1 == k2;
}
When Clojure calls a function with such signature, it has to box the primitive values in the arguments. Doubles are boxed by calling Double.valueOf(double). The unusual equality semantics of NaN make the examples confusing, but in every case, identical? compares the reference equality of boxed values, not numerical equality.
By looking at disassembled byte code (see below), we can see what's happening. Here's summary.
- In examples 1, 2, and 4, each of the arguments to identical?is boxed individually.
- In example 3, ##NaNis boxed only once and the same boxed value is used as the both arguments ofidentical?.
Here are a few more examples. Note that the behavior of ##Inf matches that of ##NaN.
(identical? ##Inf ##Inf)          ; true
(let [x ##Inf] (identical? x x))  ; false
(identical? 1.0 1.0)              ; false
(let [x 1.0] (identical? x x))    ; false
(def my-one 1.0)
(identical? my-one my-one)        ; true
Conclusion
Everything works as it should. My mental model about identical? was wrong. It's more like (Object)... == (Object) ... than plain ==.
Lesson learned:  Use identical? only for reference equality.
Disassembled examples
The code has been disassembled with clj-java-decompiler.
Example 1 (identical? Double/NaN Double/NaN)
class user$fn_line_1__254
    Minor version: 0
    Major version: 52
    Flags: PUBLIC, FINAL, SUPER
    
    public void <init>();
        Flags: PUBLIC
        Code:
                  linenumber      1
               0: aload_0        
               1: invokespecial   clojure/lang/AFunction.<init>:()V
               4: return         
    
    public static java.lang.Object invokeStatic();
        Flags: PUBLIC, STATIC
        Code:
                  linenumber      1
               0: getstatic       java/lang/Double.NaN:D
               3: invokestatic    java/lang/Double.valueOf:(D)Ljava/lang/Double;
                  linenumber      1
               6: getstatic       java/lang/Double.NaN:D
               9: invokestatic    java/lang/Double.valueOf:(D)Ljava/lang/Double;
                  linenumber      1
              12: invokestatic    clojure/lang/Util.identical:(Ljava/lang/Object;Ljava/lang/Object;)Z
              15: ifeq            24
              18: getstatic       java/lang/Boolean.TRUE:Ljava/lang/Boolean;
              21: goto            27
              24: getstatic       java/lang/Boolean.FALSE:Ljava/lang/Boolean;
              27: areturn        
        StackMapTable: 00 02 18 42 07 00 1D
    
    public java.lang.Object invoke();
        Flags: PUBLIC
        Code:
                  linenumber      1
               0: invokestatic    user$fn_line_1__254.invokeStatic:()Ljava/lang/Object;
               3: areturn        
    
    static {};
        Flags: PUBLIC, STATIC
        Code:
                  linenumber      1
               0: return         
Example 2 (let [x Double/NaN] (identical? x x))
Skipped; similar to example 4.
Example 3 (identical? ##NaN ##NaN)
Note that the boxing happens in the static block.
class user$fn_line_1__246
    Minor version: 0
    Major version: 52
    Flags: PUBLIC, FINAL, SUPER
    
    public static final java.lang.Object const__1;
        Flags: PUBLIC, STATIC, FINAL
    
    public void <init>();
        Flags: PUBLIC
        Code:
                  linenumber      1
               0: aload_0        
               1: invokespecial   clojure/lang/AFunction.<init>:()V
               4: return         
    
    public static java.lang.Object invokeStatic();
        Flags: PUBLIC, STATIC
        Code:
                  linenumber      1
               0: getstatic       user$fn_line_1__246.const__1:Ljava/lang/Object;
               3: getstatic       user$fn_line_1__246.const__1:Ljava/lang/Object;
                  linenumber      1
               6: invokestatic    clojure/lang/Util.identical:(Ljava/lang/Object;Ljava/lang/Object;)Z
               9: ifeq            18
              12: getstatic       java/lang/Boolean.TRUE:Ljava/lang/Boolean;
              15: goto            21
              18: getstatic       java/lang/Boolean.FALSE:Ljava/lang/Boolean;
              21: areturn        
        StackMapTable: 00 02 12 42 07 00 17
    
    public java.lang.Object invoke();
        Flags: PUBLIC
        Code:
                  linenumber      1
               0: invokestatic    user$fn_line_1__246.invokeStatic:()Ljava/lang/Object;
               3: areturn        
    
    static {};
        Flags: PUBLIC, STATIC
        Code:
                  linenumber      1
               0: ldc2_w          NaN
               3: invokestatic    java/lang/Double.valueOf:(D)Ljava/lang/Double;
               6: putstatic       user$fn_line_1__246.const__1:Ljava/lang/Object;
               9: return         
Example 4 (let [x ##NaN] (identical? x x))
Note two Double.valueOf calls in the invokeStatic method.
class user$fn_line_1__250
    Minor version: 0
    Major version: 52
    Flags: PUBLIC, FINAL, SUPER
    
    public void <init>();
        Flags: PUBLIC
        Code:
                  linenumber      1
               0: aload_0        
               1: invokespecial   clojure/lang/AFunction.<init>:()V
               4: return         
    
    public static java.lang.Object invokeStatic();
        Flags: PUBLIC, STATIC
        Code:
                  linenumber      1
               0: ldc2_w          NaN
               3: dstore_0        /* x */
               4: dload_0         /* x */
               5: invokestatic    java/lang/Double.valueOf:(D)Ljava/lang/Double;
               8: dload_0         /* x */
               9: invokestatic    java/lang/Double.valueOf:(D)Ljava/lang/Double;
                  linenumber      1
              12: invokestatic    clojure/lang/Util.identical:(Ljava/lang/Object;Ljava/lang/Object;)Z
              15: ifeq            24
              18: getstatic       java/lang/Boolean.TRUE:Ljava/lang/Boolean;
              21: goto            27
              24: getstatic       java/lang/Boolean.FALSE:Ljava/lang/Boolean;
              27: areturn        
        StackMapTable: 00 02 FC 00 18 03 42 07 00 1B
    
    public java.lang.Object invoke();
        Flags: PUBLIC
        Code:
                  linenumber      1
               0: invokestatic    user$fn_line_1__250.invokeStatic:()Ljava/lang/Object;
               3: areturn        
    
    static {};
        Flags: PUBLIC, STATIC
        Code:
                  linenumber      1
               0: return