What I get from the sentence is "he didn't occupy the office but instead he wear it" as to wear glasses, to emphasize he barely fits in it. Is it correct? And if so can you explain the grammar rule because I don't understand how "as wear it" works in this sentence.
No, they don't have the same meaning, although they are usually somewhat related.
Go into mourning means that you enter into a state of grief; it describes a condition of the heart. If I begin to feel sorrow and pain at the loss of a loved one, then I have gone into a state of mourning.
Wear mourning, on the other hand, means putting on clothes that display an outward expression of that grief. It is more of a symbolic action.
That all said, the two normally go hand-in-hand. You generally wear mourning because you have gone into mourning, although there may be some exceptions to that (such as a widow who has murdered her husband, but wants to look as though she is grieving in order to allay suspicion).
A Google books search on the expression wear mourning reveals some interesting customs across different cultures. For example, some people are (or were) recommended to "wear mourning for a year." The fact that such a predetermined length of time could be prescribed illustrates how there is not always an exact correspondence between the true inward feelings of mourning, and the symbolic act of wearing mourning clothes.
In your linked page, the examples are like
[subject] wearing mourning dress...
This usage is OK since wearing is the verb and mourning dress is the object.
go into mourning means the person starting the process of mourning, as in:
He will go into mourning.
though go could be conjugated like
He is going into mourning.
As for wear mourning, many examples are like "they would wear morning clothes" (here wear is a verb rather than noun or part of a noun phrase), but some were like that used in this Wikipedia link Mourning which has related historical information
Parents would wear mourning for a child for "as long as they feel so disposed".
It seems to be mainly of UK usage. I have never heard it used this way. wear needs to be the verb, so I guess mourning is the object meaning mourning clothes.