Learn English – Are the words “network packet” and “network package” interchangeable

I think you mostly have it right, but for further exemplification, here are the most common immediately following collocates for each these words from the COCA:

Most common collocates for mandatory

1   TESTING      151   
2   MINIMUM      125   
3   RETIREMENT   107   
4   MINIMUMS     100   
5   REPORTING    87    
6   SENTENCES    78    
7   EVACUATION   74    
8   SENTENCING   74    
9   DISCLOSURE   63    
10  ARBITRATION  56    
11  SPENDING     41    
12  DRUG         36    
13  SENTENCE     35    
14  EVACUATIONS  32    
15  ARREST       27    
16  PRISON       27    
17  COUNSELING   26    
18  OVERTIME     24    
19  RECYCLING    23    
20  BUSING       22    
21  SANCTIONS    22    
22  INSURANCE    21    
23  SERVICE      21    
24  HIV          20    
25  RELEASE      20    

Most common collocates for compulsory

1   EDUCATION       86
2   MILITARY        53
3   SCHOOL          31
4   SCHOOLING       21
5   STERILIZATION   19
6   LICENSING       18
7   ATTENDANCE      18
8   NATIONAL        16
9   SERVICE         15
10  MILITIA         14
11  MEMBERSHIP      14
12  JURISDICTION    11
13  HEALTH          11
14  HETEROSEXUALITY 10
15  LABOR           10
16  ARBITRATION     9 
17  LICENSE         9 
18  FIGURES         9 
19  PROCESS         9 
20  PUBLIC          9 
21  RETIREMENT      8 
22  DISPUTE         7 
23  ARMY            7 
24  VOTING          6 
25  INSURANCE       6 

Most common collocates for obligatory

1   SERVICE          10 
2   NATURE           8  
3   STOP             8  
4   REFERENCE        6  
5   STRUCTURE        5  
6   CELIBACY         4  
7   EXCHANGE         4  
8   VISIT            4  
9   READING          4  
10  MILITARY         4  
11  QUESTION         4  
12  SLAP             3  
13  NOD              3  
14  RETURNS          3  
15  BASIC            3  
16  PERIOD           3  
17  UPON             3  
18  PHOTO            3  
19  BOOK             3  
20  EARLY            3  
21  PARTHENOGENESIS  2  
22  PREPOSITION      2  
23  PILGRIMAGES      2  
24  HAGGLING         2  
25  ALTRUISTIC       2  

As you can see, mandatory and compulsory have much stronger collocates. The collocates for mandatory are more things that are simply required by law, policy, or rule, whereas compulsory is more for specifically things that someone must do (e.g. education, military, sterilization). Obligatory, on the other hand, doesn’t have much of a strong affinity for certain things, and can be used generally for anything that is required by social custom.

Interesting question.

There really aren't any common words that express the difference between equality and equivalence. Wearing the same hat, eating the same food, driving the same car — all of these things point to equivalence rather than equality or identity.

What the hat, food and car represent here are instances of classes, but not the same instances. To express that one instance of car is identical to another instance — for example, that you and I were driving the same Ford Fusion, California License Plate No. FOOBAR1 (sorry if that is a real plate number), on the same day, I in the morning and you in the afternoon — we would have to go out of our way to express that by actually citing the plate number or explaining that I loaned you my car or you loaned me yours.

Even to say we were driving the identical car would not cause the listener, at first, to suspect we meant the exact same car with the same plate number (and serial number). Identical here would be understood only to mean we were driving the same make, model, year, and color vehicle. Even saying "the exact same" car would still be understood to mean a car exactly like the other car, not the car itself.

Look at NOAD's list of synonyms for identical:

identical adjective 1 wearing identical badges: indistinguishable, (exactly) the same, uniform, twin, duplicate, interchangeable, synonymous, undifferentiated, equivalent, homogeneous, of a piece, cut from the same cloth; alike, like, matching, like (two) peas in a pod; similar.

Not one of those synonyms expresses anything like the Law of Identity (A = A) in mathematics or the strict equality operator in some programming languages (=== instead of ==), even though the root of the word identical is, in fact, the same as for identity: Latin idem meaning "the same".

Even when we speak of things that point to identity, such as fingerprints or DNA, saying that a sample of DNA is identical to the DNA found at a crime scene does not mean the strands are the same strands, but that they come from the same person.