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I am not sure whether there's a clinical term for it but if you can help me find the closest term for such behaviour I can do some research on it.

What do you call a person that disagrees with your statement but then is effectively saying the same thing?

NOTE: The person in question has been exposed to very simple questions that can't be classified as complex and yet failed to notice what he/she was doing there.

I will explain this in practical terms using one example (this example is not the same the person was exposed to)

I: The football match is on Wednesday (saying this Sunday)
He/she: No it's not, it's in 3 days. (which effectively is Wednesday)

I know the example is a bit late but it's to give you the gist of the situation I'm trying to explain.

A person who argues for the sake of arguing, who prefers controversy and dispute to discussion and agreement, may be called eristic. A more modern term which could be used in this context is denier. Of course, this type of person could also be called disagreeable.

I have often heard people say "We're vehemently agreeing with each other."
vehement:
(1) zealous; ardent; impassioned:
(2) characterized by rancor or anger; violent:

http://dictionary.reference.com/browse/vehemently

Another common phrase is "You're preaching to the choir" referring to a pastor directing his sermon to the people in the building who most agree with him.

I see two branches to this question.

Branch one assumes that he/she is aware that there is a technical agreement but are continuing to argue for the sake of argument.
Possibilities are:

1. Imperative, Domineering - (They just want to win) It appears from synonym descriptions that domineering is more accurate if it's a decision-making argument, but imperative is better as a general-purpose word.


2. Ornery, Combative, Argumentative, Eristic- (They just want to argue)

Branch two assumes that the other person is unaware of the agreement already reached

1. 
   Dogmatic - I think that this one may be the best fit, since many of the examples given mention the futility of arguing with a dogmatic individual. Dogmatic is particularly appropriate if the disagreement is over politics, religion, etc.


2. Dogged and Dense - requires two words to get the full connotation in print, but in speech either word would suffice

*I have only linked two words, but that's because of rep limits, not laziness

I'm not too sure about this but it this may be termed as hypocritical or dogmatic beahaviour, meaning:

hypocritical

behaving in a way that suggests one has higher standards or more noble beliefs than is the case.

dogmatic

expressing personal opinions or beliefs as if they are certainly correct and cannot be doubted

In the case of your example, the point brought forth is similar (that the football match occurs on Wednesday) but the other party simply rejects the opposing idea probably believing that he has an answer that is 'more' or 'certainly' correct, of 'higher standard' than the one given or just simple believing the fact that he 'cannot be doubted'.

I would suggest the phrase "re-contextualizing."

Basically, the person engaging in this behavior is someone who has a very rigid frame of reference or worldview, and is unable to perform simple abstraction of your frame of reference (in the example cited, calibrated in day-name) to their frame (calibrated in days hence).