The line
zoobis = zoo;
just makes both the zoobis and the zoo variables point at the same object. Then, the line
zoobis.animals = zoo.animals.filter((animal) => animal.color === "brown");
modifies the state of the one object both variables are pointing at, replacing its animals property with a new filtered array.
Naturally, you see the same object state regardless of whether you look at it via zoobis or via zoo, as both of those variables refer to the same object.
Some ASCII-art:
Once you've created the zoo object, you have something vaguely like this in memory:
                                                                          +−−−−−−−−−−−−−−−−+                    
                                                                       +−>|    (object)    |                    
                     +−−−−−−−−−−−−−−−−−−−−−−−−−−−+                     |  +−−−−−−−−−−−−−−−−+                    
zoo:[Ref71234]−−−−−−>|         (object)          |                     |  | id: 1          |                    
                     +−−−−−−−−−−−−−−−−−−−−−−−−−−−+                     |  | name: "Foufou" |                    
                     | name: "Valley of monkeys" |   +−−−−−−−−−−−−−−−+ |  | color: "brown" |                    
                     | animals: [Ref55412]       |−−>|    (array)    | |  +−−−−−−−−−−−−−−−−+                    
                     +−−−−−−−−−−−−−−−−−−−−−−−−−−−+   +−−−−−−−−−−−−−−−+ |                                        
                                                     | 0: [Ref45132] |−+                      +−−−−−−−−−−−−−−−−+
                                                     | 1: [Ref45174] |−−−−−−−−−−−−−−−−−−−−−−−>|    (object)    |
                                                     | 2: [Ref45228] |−+                      +−−−−−−−−−−−−−−−−+
                                                     +−−−−−−−−−−−−−−−+ |  +−−−−−−−−−−−−−−−−+  | id: 2          |
                                                                       +−>|    (object)    |  | name: "Toutou" |
                                                                          +−−−−−−−−−−−−−−−−+  | color: "brown" |
                                                                          | id: 3          |  +−−−−−−−−−−−−−−−−+
                                                                          | name: "Moumou" |                    
                                                                          | color: "blue"  |                    
                                                                          +−−−−−−−−−−−−−−−−+                    
Notice that zoo contains a reference to the object (conceptually shown above as [Ref71234], but you never actually see the value of an object reference in code).
Then after zoobis = zoo; you have:
zoo:[Ref71234]−−+
                |                                                         +−−−−−−−−−−−−−−−−+                    
                |                                                      +−>|    (object)    |                    
                |    +−−−−−−−−−−−−−−−−−−−−−−−−−−−+                     |  +−−−−−−−−−−−−−−−−+                    
                +−−−>|         (object)          |                     |  | id: 1          |                    
                |    +−−−−−−−−−−−−−−−−−−−−−−−−−−−+                     |  | name: "Foufou" |                    
                |    | name: "Valley of monkeys" |   +−−−−−−−−−−−−−−−+ |  | color: "brown" |                    
                |    | animals: [Ref55412]       |−−>|    (array)    | |  +−−−−−−−−−−−−−−−−+                    
zoo:[Ref71234]−−+    +−−−−−−−−−−−−−−−−−−−−−−−−−−−+   +−−−−−−−−−−−−−−−+ |                                        
                                                     | 0: [Ref45132] |−+                      +−−−−−−−−−−−−−−−−+
                                                     | 1: [Ref45174] |−−−−−−−−−−−−−−−−−−−−−−−>|    (object)    |
                                                     | 2: [Ref45228] |−+                      +−−−−−−−−−−−−−−−−+
                                                     +−−−−−−−−−−−−−−−+ |  +−−−−−−−−−−−−−−−−+  | id: 2          |
                                                                       +−>|    (object)    |  | name: "Toutou" |
                                                                          +−−−−−−−−−−−−−−−−+  | color: "brown" |
                                                                          | id: 3          |  +−−−−−−−−−−−−−−−−+
                                                                          | name: "Moumou" |                    
                                                                          | color: "blue"  |                    
                                                                          +−−−−−−−−−−−−−−−−+                    
Notice how the value of zoo ([Ref71234]) was copied into zoobis, but that value is just a reference to the one object.
Then after [filter]:
zoo:[Ref71234]−−+
                |                                                         +−−−−−−−−−−−−−−−−+                    
                |                                                      +−>|    (object)    |                    
                |    +−−−−−−−−−−−−−−−−−−−−−−−−−−−+                     |  +−−−−−−−−−−−−−−−−+                    
                +−−−>|         (object)          |                     |  | id: 1          |                    
                |    +−−−−−−−−−−−−−−−−−−−−−−−−−−−+                     |  | name: "Foufou" |                    
                |    | name: "Valley of monkeys" |   +−−−−−−−−−−−−−−−+ |  | color: "brown" |                    
                |    | animals: [Ref65241]       |−−>|    (array)    | |  +−−−−−−−−−−−−−−−−+                    
zoo:[Ref71234]−−+    +−−−−−−−−−−−−−−−−−−−−−−−−−−−+   +−−−−−−−−−−−−−−−+ |                                        
                                                     | 0: [Ref45132] |−+                      +−−−−−−−−−−−−−−−−+
                                                     | 1: [Ref45174] |−−−−−−−−−−−−−−−−−−−−−−−>|    (object)    |
                                                     +−−−−−−−−−−−−−−−+                        +−−−−−−−−−−−−−−−−+
                                                                                              | id: 2          |
                                                                                              | name: "Toutou" |
                                                                                              | color: "brown" |
                                                                                              +−−−−−−−−−−−−−−−−+
Notice how you replaced the old animals value ([Ref55412]) with a new value ([Ref65241]) because you created and stored a new array.
If you want to copy zoo, you can use Object.assign or property spread (ES2018+):
zoobis = Object.assign({}, zoo);
// or
zoobis = {...zoo};
That makes a shallow copy (so for instance, both objects' animals property points to the same array). A shallow copy is sufficient for what you've shown (but might be problematic if you have other properties on zoo referring to objects).
If you want to make a deep copy, see this question's answers.