"I thank you, Ivan Ivanovich."
Translation:Благодарю вас, Иван Иванович.
That's "thank you, Ivan Ivanovich", the point of the sentence is to say that someone is explicitly saying thanks to that certain person. The 'thank you' phrase lost some of the initial meaning and the "[I] thank [you]" does not sound as formal. In Russian the whole form has stronger meaning than everyday "than you".
I believe "благодарю" can be regarded as a set expression (if the word "expression" can be applied to one word), the same way as "thank you" is used without "I". Yes, saying "I thank you" adds to the level of formality, but somehow that doesn’t work with "Благодарю вас".
BTW, "благодарю" can really be used as one word, without "вас". (This way it sounds less formal.)
No. This concept will be easier to understand if you have cases in your native language but even if you don't, you'll get the hang of it eventually. In "Благодарю вас" you is the object so it's in the accusative. In "Спасибо вам" however, where спасибо is not a verb, it says вам (dative) as in "(I give) thanks to you"