Under the formal rules of English grammar, a singular subject requires a singular verb.
Because of a programming error, the identity number of John and Mary is the same.
One identity number requires a singular verb.
Of course, the identity numbers of John and Mary are different.
Two identity numbers require a plural verb.
The meaning of "lovely" and "wonderful" is the same
is grammatically correct.
And also correct grammatically is
The meanings of "lovely" and "wonderful" are very close to the same.
Also correct grammatically is
The meaning of "lovely" and "wonderful" is close to the same.
"Meaning" requires "is" under the rules of English grammar. Of course, the sentence, however grammatical, is nonsense. If the meanings are not the same, then it makes no sense to label them as a single meaning.
You can be nonsensical in perfect grammar, and logical in incorrect grammar. Do not confuse grammar and sense.
Probably what was in the tutor's mind was something like
In certain usages, the words have an identical meaning, but in others they do not.
When we have complex thoughts, sometimes the grammar gets tangled and sometimes the thought itself.