Both 243's and Gangnus's answers are correct, enough that I may not have to add anything. However, after trying to put myself in the OP's shoes, I can see that the confusion seem to arise from a few causes, and I believe that some angles of the OP's confusion haven't been addressed yet.
I would like to break the OP's confusion down into these following questions:
- What is the function of the best/the most/all the same?
(as indicated by the OP's "What kind of structure are they?")
- Does the use of the word the make them a noun phrase?
(as indicated by the OP's "the word 'the' should be a noun phrase because 'the + adjective' actually means 'the + adjective + noun'")
- Shouldn't the best (or the most and all the same) be a noun phrase, because it seems that the OP have seen the best being used as a noun phrase before, and it might seem to mean the best (someone or something)?
(In the OP's words, "So I wonder: all the same = all the + same + (?), the best = the best + (?)".)
- What do the best and the most mean?
(This might sound a little strange, because the OP seems to know the meaning. However, reading the question carefully and I came to realize that the OP couldn't tell whether the best functions as an adjective or as an adverb.)
- What does this all the same mean?
(This, actually, deserves to be another question. However, the Op seems to be tricked to think that they should be considered together at once, due to the similar use of the.)
I will answer these questions, one at a time.
(a) What is the function of 'the best', 'the most', and 'all the same'?
They are 'adverbials', also known as 'adverbial phrases' or 'adverb phrases'.
I think the OP's greatest confusion is likely a result of the fact that both best and most can be the superlative forms of either an adjective or and adverb.
One fact that most learners are already aware of, but might not be able to recall when reading a text is English has degrees of comparison applied to both adjectives and adverbs. The "comparative" indicates greater degree (e.g. bigger, more fully). The "superlative" indicates greatest degree (e.g. biggest, most fully). The tricky thing about best and most is that, they are the superlative forms of the two adjectives, good and much (or many),
good -> better (comparative) -> best (superlative)
many or much -> more (comparative) -> most (superlative)
and they also are the superlative forms of the adverbs, well and much:
well -> better (comparative) -> best (superlative)
much -> more (comparative) -> most (superlative)
To refresh the use of comparatives and superlatives for adjectives and adverbs, let's consider some simple examples. Let's consider the adjective tall:
John is tall. John is taller than James. John is the tallest boy in our class.
Who is the tallest? -- John is the tallest.
Note the use of the definitive article 'the' before a superlative.
Let's consider the adverb hard:
They worked hard.
They worked harder, they were more honest.
The old people work hardest.
Note that the use of 'the' with superlatives is optional, e.g.
His shoulders hurt the worst.
Let's get back to the OP's question:
I like it the best.
I like it the most.
To test if they function as adjectives or as adverbs, we can try substitute them with their plain forms: good vs. well, and many vs. much.
I like it the best.
I like it well (enough).
I like it many.
I like it (very) much.
That should make it clear that they function as adverbs, that is they are 'adverb phrases'. The phrase "all the same" is also an 'adverb phrase', but it might be better to discuss about it later.
(b) Does the use of the word 'the' make them a noun phrase?
Not always.
Recall one of our examples:
Who is the tallest? -- John is the tallest.
Asking about the quality of the tallest doesn't make this "the tallest" a noun phrase. Compare:
Who are tall? -- John, James, and Jimmy are (tall).
Also consider these examples,
The older I get, the happier I am.
The more difficult it is, the more I like it.
The more expensive gasoline becomes, the less people drive.
None of them are a noun phrase.
(c) Shouldn't 'the best' ('the most', 'all the same') be a noun phrase?
In some sentences, the best (also the most, the same, the better, the more) is used as a noun phrase. In others, it's not. Here are the examples showing the sentences they are used as a noun phrase: (NOTE: Some references classify them as a pronoun rather than a noun phrase.)
We all ate a lot, but Ashley ate the most. ("The most" is what Ashley ate.)
I want the best for my family. ("The best" is what I want.)
I would do the same if I had the chance. ("The same" is what I would do.)
She picked the better of two choices. ("The better" is what she picked.)
The more was contributed towards the building the more magnificent it would be. ("The more" was contributed.)
However, just as like those examples in (b), they can be used as adjective phrases or adverb phrases too. Consider these examples:
Direct export works the best if the volumes are small. ("The best" is how direct export works.)
Of all the possible guests out there, who is the one you want the most? - Tom Cruise (is the one I want the most to be the guest on this show). ("The most" is how I want Tom Cruise.)
(d) What do 'the best' and 'the most' in the example mean?
In "I like it (the) best," and "I like it (the) most," both the best and the most are used as superlative adverbs of degree, meaning: to the greatest extent, extremely, very.
Also note that the word the is optional.
(e) What does 'all the same' mean?
Dictionaries usually define all the same as a phrase, meaning "despite what has just been said". (Thanks go to learner's comment below.)
However, if that definition doesn't work well for you, you can simply read it (roughly) as "though". For example, "He did it all the same" ~ "He (still) did it though". -- Students' English Grammar by Jake Allsop lists it under entry 12.20 Adverbials which join or link ideas and sentences, with a note, "..., though usually means all the same or on the other hand."
