1) So far Wie has been interpreted as 'how'. For example
Wie geht es? How are you?
Wie viele tagen? How many days?
2) It can also be used as a replacement for the second 'as'. For example
so blau so (as blue as) is incorrect so blau wie is correct.
so schön wie (as beautiful as)
3) Wie can be used instead of 'than' as well. Especially when both things being compared are considered equal. For example,
Ich bin großer wie du (I am taller than you) is incorrect Ich bin so groß wie du_is correct
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After als and wie in comparisons, the part after it has the same case as the thing you are comparing to.
Here, you are comparing the doctor to your son.
The doctor is the subject of ist and so it's in the nominative case -- therefore, mein Sohn also has to be in the nominative case.
My guess would be that the sentence does not reffer to the height of either person relative to anyone else, meaning that we do not know if the the people are relatively tall, as is implied in the phrase "so tall"; it could be that both the son and the doctor a very short. The sentence does not seem to specify. It only says that the two people are of similar height.
Nope, that doesn't make any sense in English. (I'm not even sure I would understand that if it was said to me.)
It looks like you've just taken the standard dictionary definitions for each individual word ("so" = "so," "wie" = "like"), and you just can't do that "direct" translation here. You need to translate into correct English phrasing, which would be "My doctor is as tall as my son."
For the most part, it's a pretty word-for-word sentence. "Meine Ärztin" = "My doctor" / "ist" = "is" / "so [adj.] wie" = "as [adj.] as" / "groß" = "big" / "mein Sohn" = "my son."
The tricky parts are probably the "so ... wie," which is simply German's phrasing for "as ... as," and the fact that "mein Sohn" is nominative, which is because "wie" isn't a preposition but an adverb, so "mein Sohn" takes the same case as "meine Ärztin," i.e. nominative.