An approximate answer is one which is close to the correct answer. Likewise, we can talk of an approximate model, or approximate methods in mathematics.
The etymology is from the Latin ad, "to" and proximus, "closest", see e.g. here. I don't think this can be disputed. Occasionally proximate is used as a synonym, though that word has a different meaning in legal English.
So if an approximate answer is close, a very approximate answer should be very close.
In practice, "very approximate" or "very approximately" is overwhelmingly used with the opposite meaning of "roughly", indicating that an answer or estimate is not necessarily accurate. For examples, search Google.
Is there any justification for this use? Is there any justification for my suggested use? Should we avoid the phrase altogether?