Let's start with what all linguists seem to agree on.
The following sentences are fully grammatical in most standard variants of English:
- She is smarter than me.
- He is as tall as me.
- He is sincere, just like me.
(This is just one of many weird rules that non-native speakers have to be taught systematically, with drills.)
In most if not all natural variants of English, the following alternatives do not occur naturally but are generally considered grammatical and in fact prestigious, due to the efforts of prescriptivists:
- She is smarter than I.
- He is as tall as I.
For a discussion of this me vs. I phenomenon, see the Language Log article Patterns of prestigious deviance. It has plenty of pointers to scholarly literature. I'll try to get a convincing explanation out of the chaotic findings and some additional background knowledge.
Some languages use personal pronouns, which may occur in several forms depending on case. E.g. for the English first person singular pronoun we have the forms I (subject case) vs. me (object case) vs. my (possessive; formerly genitive case) vs. myself (reflexive). Some languages mark person and number on the verbs. English has only a rudiment of this in the third person singular marker -s. Latin has a full system. In spoken French the distinctions are mostly lost, but in written French most still exist.
Languages appear to oscillate between the two options. We can see this clearly in French. Classical Latin worked without personal pronouns just with verb endings. Personal pronouns such as ego were only ever used when they were stressed. But this changed. In later forms of Latin, the verb endings became less distinctive and the personal pronouns were used increasingly even when they were not stressed. Ultimately, stressed ego evolved into French je, which is obligatory and cannot be stressed! This unstressed pronoun has already started the process of merging into the following verb. It began with contractions when a verb starts with a vowel (as in j'ai), but nowadays in spoken French contractions become increasingly common even before consonants (as in j'suis, often pronounced juis). The result will be a system in which person and number are marked at the beginning of a verb and the old pronouns aren't really pronouns any more. It should be clear that at that point French will need new pronouns, and in fact they already exist! They are known as emphatic pronouns. The emphatic first person singular pronoun is moi, and it is derived from me, the accusative form of Latin ego.
From this we learn that in French, the language of British nobility over centuries, there is a close connection between the object case of the first person singular pronoun and its emphatic form, and that the object cases in English and French are very similar (me vs. moi, with the latter derived from Latin me).
Now we come to a second piece in the jigsaw puzzle: The evolution of subject case I. In Proto-Germanic, this pronoun had the forms ik and ek, which is remarkably close to ego. In many Germanic languages it is still ik or something very similar even today. (E.g. in Danish it's jeg, which is also close to French je.) But in some Germanic languages and dialects, it was shortened to just i (pronounced like ik without the k, not as a diphthong!). E.g., this is the case in modern Bavarian dialects of German. But it was also the case once in English. In other words, the first person singular pronoun almost vanished and so was no longer suitable for receiving emphasis. Just like in French!
One strategy for dealing with this problem is to add an extra element to the pronunciation to make the word longer so that it can be emphasised. English followed this strategy, and so the modern pronunciation of I is with a diphthong. (Similar strategies are still productive in German, e.g. nein can become nei-en for emphasis. I don't know if English speakers still use them.)
But apparently, at the time when I was still pronounced as a single vowel, people also used the French strategy when they wanted to stress it: They used the object case me instead. If English had no emphatic pronouns, we would expect the subject case in single-word responses to the question "Who is there?". But what we actually get is, in the overwhelming majority of cases, not I but me. Just like the French response is never je and always moi.
Since grammar is always in a state of change, it is rarely the case that a language has, or lacks, a certain feature 100 %. But a good case can be made that English largely has emphatic pronouns analogous to those of French. The system is a bit more chaotic than the French one; maybe this is due to interference by prescriptivists at a time when the system wasn't fully developed yet.
Note that this is just one plausible hypothesis to fit the facts as I found them. It is also possible that the single-vowel pronunciation of I always co-existed with the variant ik before it turned into the modern diphthong pronunciation. In that case it appears that a stronger French influence must be postulated to explain the modern uses of me.
I should also note that Bavarian has the single-vowel first-person singular i today, and that in Bavarian it can receive emphasis. So it seems likely that the French example must have played a role either way.
Now what's wrong with the following?
- He is sincere, just like I.
In most natural varieties of English, this is simply not grammatical. The likely reason is that in an earlier stage of the language, I was a single short vowel that was barely audible and could only be used when it attached to the following word. So, under French influence and largely for the same reasons as French speakers, English speakers used the object case form instead.
The reason to use this ungrammatical form anyway would be that it is part of an elevated, formal register which was to some extent artificially constructed by prescriptivists. However, this sentence essentially has a stream-of-consciousness pattern with a main thought followed by an afterthought. Sentences following this pattern aren't really part of the formal register in question. Therefore, regardless of whether we use I or me, you wouldn't expect this sentence in close proximity to a sentence such as It is I.
So the problem with this sentence is that it is incongruous in a way that makes it hard to choose between applying the natural rules of grammar (demanding me) and applying the artificial rules spread by parts of the educational system in English-speaking countries (demanding I).
By the way, this is an area in which ESL speakers seem to be at an advantage. Most are not subjected to the ideology of English prescriptivism, and even if they are - as they vaguely remember hours of drills for using me where I would appear more logical, it is much harder to confuse them than it is to confuse native speakers. Until they learn about the formal register, many will even interpret constructions such as It is I as typical mistakes of fellow non-native speakers! Also, a typical ESL speaker is influenced more by the classical English and American authors (most of whom aren't under the prescriptivists' spell) and less by the writings of random authors who follow bad language advice.