You seem to adress several issues at once. A difference in glyph design traditions between Bulgaria and Russia may be one of them. But I would not call this “identity crisis”.

If I understand you properly, you are concerned with glyphic detailing which, e.g., makes the Cyrillic K in a font looking different from the Latin K or the Greek Kappa. – Am I right?
Well, these things have been discussed many times already as I recall. What you label as “very strange and obscure things” is perhaps just a certain convention, one of several in existence. Then the font designer has to make decisions, tricky ones, sometimes. But such issues belong to type in general, to multi-scriptiv type in particular and are not peculiar for the Cyrillic branch alone.
I myself decided to just issue Andron Cyrillic in different variants. Others implement glyph variation options via Opentype features …

I’d welcome you to give us some insights on your personal preferences for (Bulgarian?) Cyrillic.

I like it when Cyrillic and Latin look very similar, but if I were to make a font with very Latin looking Cyrillic letters I think I'd be criticized for not understanding the subtleties of Cyrillic. I'd imagine that designers whose first alphabet is not Cyrillic (or Greek) are under a good deal of pressure to include all sorts of regional details in order to prove they're qualified to work with those alphabets in the first place. Better to include lots of unnecessary subtle changes than to leave out a few important ones.

It's funny you mention k— in Russian, at least, the arm of the k pretty consistently points down or forward, whereas Latin k arms are mostly straight or curved up. That particular detail is one I'd be hesitant to leave out were I to design my own Cyrillic face (perhaps a Bulgarian k is not quite like a Russian k?).

Oddly enough, just yesterday I replaced the old looped k in my little cursive font with (what seems to me) a Russian-looking forward-pointing k. I kinda like it and intend to stick with it...

I'm a Canadian with no knowledge of Cyrillic prior to my typographic work.

In developing the Modern Suite, released in 2008, I decided to include Bulgarian alternates (as well as the Big Yus character), prompted by a Maxim Zhukov seminar on Cyrillics organized by the TDC in New York, and based on my study of types published by the Hermes foundry, and discussions here at Typophile.

These Modern Suite alternates are not designated as the default for text identified as being set in Bulgarian (BGR), but are accessible as an OpenType stylistic set. That seemed like the "best practice". AFAIK (I have 10 distributors, so aren't always up to date on who is buying my fonts), nobody in Bulgaria has licensed any Modern Suite fonts yet, so the Bulgarian alternates appear to be a waste of time--although I haven't made any efforts to promote the fonts in Bulgaria.

TypeTogether's Skolar also includes Bulgarian alternates.

BTW, we have had a similar situation with the Latin script, where the single bowl version of "g" may be considered more European than the double bowl version, which tends to be preferred in North American sans serifs. At least, during most of the 20th century.

Letters like "K", "k" which even sound alike in all alphabets have in fact different faces.

Because they have different histories. It seems to me that ignoring different histories is more likely to result in genuine identity crises than worrying that letters in different writing systems and different styles look, well, different.

The "crisis" is also inherent to the evolution of language.
The law of vowel-change was discovered by Rasmus Rask c.1820.
In the Latin script, the related pronunciation of "v", "w" and "q" has varied across the centuries and from country to country.
Even today, a language like English is pronounced quite differently in different parts of the countries where it is spoke.
So why should alphabets conform?

As various sages have opined, "Never trust a man who only knows one way to spell a word".
Or one way to glyph a character.

If I were to try to take the Greek alphabet, and modify the lowercase of it so that when the uppercase Greek letter looks like an uppercase Latin letter, then the lowercase letter matches again - and create new lowercase forms, then, for the other Greek letters to "fit in"... people would laugh at me, and certainly the Greeks would never use such a thing.

Yet, there are different forms for the capital Greek letter Upsilon, and Greeks are more likely than Latin-alphabet users to use a form that looks like the Latin Y.

I think you are asking a legitimate question, even though what you seem to be asking for sounded like my silly example of making Greek conform to Latin. The Greek and Latin alphabets developed naturally over a long period of time. The Cyrillic alphabet is of more recent origin, and perhaps the current form used in Russia and Bulgaria is an arbitrary and artificial modification, instead of a natural development, from the larger Slavonic alphabet.

Greek has its own problems. The "Porson" style of Greek typeface, most familiar to Latin alphabet users, mixes a cursive, sloped lowercase with upright Roman-like capitals. This is very handy when typesetting mathematics in English, but it isn't well suited to normal text composition. The Greeks use other styles, therefore, such as sans-serifs, but the fact that there is no really Hellenic serifed style to standardize on as the basis for Greek typefaces is a problem.

The Latin upper and lower case is artificial too, a shotgun marriage of Roman capitals with uncials. But if someone tried to reinvent the alphabet to replace it with something more "organic", but unfamiliar, except as a curious display font (i.e. Peignot)... it would not fly.

Except perhaps for some subtle changes of peculiarities in the ordinary serifed fonts... such as stepping back from Bodoni/Didot styling... there is not much that can be done, I think, in the way of a general reform. That doesn't mean that adventurous experimentation, though, cannot be engaged in.

It occurs to me that I have perhaps misconstrued the goals of the original poster, or, at least, attempted to make a correction to his post that was irrelevant.

While I think that there is a problem with Greek letterforms due to the Porson Greek convention arising while the Greek language was suppressed under Turkish occupation, and, in fact, one might even ask if Greek needs an upper and lower case duality any more than, say, Hebrew... that may be irrelevant to the original poster's goals.

Instead, if one accepts as a postulate that the Greek alphabet in its present form is a satisfactory expression of the Greek national consciousness, on a par with the Latin alphabet for the countries that use it, then it is possible to focus on the specific question of raising the Cyrillic alphabet to parity. With a lower-case that is a mix of Latin alphabet lower-case and Latin alphabet small-capitals, this is presumably not achieved.

Once the question is framed in this way, the solution becomes obvious. For Cyrillic to raise its head with equal dignity to Greek... the reform would be to retain the existing Cyrillic upper-case, but to replace the lower-case with one based on Glagolitic letter forms.

As a compromise, the Petrine reform was a quite dignified way of moving Russia out of its backwater.
At other times, European countries have abandoned their traditional alphabet styles wholesale, in favor of the Italian model.
In particular, the blackletter, which was used in Germany, Denmark, Norway etc. until quite recently, historically speaking--much later than the Petrine reform.

Strictly speaking, the Fraktur is still part of the Latin script, but most of its alphabet is illegible to those in countries that switched to the Antiqua long ago.

"replace the lower-case with one based on Glagolitic letter forms."

Whoa, now that would be pretty radical. I'd understand if you suggested letterforms derived from cursive script, or Skoropis, or Ustav, or even Latin minuscule, but Glagolic letterforms take the cake :)

I doubt there's more then a handful of people in the world outside of Croatia who could read it.

Glagolitic is a different alphabet, not an early form of Cyrillic. The difference between the two is comparable to that between the runic alphabet and the Latin alphabet used later by Germanic peoples.

Well, I was intentionally making an extreme suggestion to poke fun at certain forms of extreme nationalism.

However, in connection with the Greek upper- and lower- case system having historical problems, I ran across Eric Gill's attempt at a reformed Greek alphabet in his Perpetua typeface in his book "Typography" the other day; this is the book with the famous quote about the history of printing being an effort to eliminate the impression that does the printing.

I had written my previous post before becoming aware of the reason why no advertiser is ever again going to use Perpetua to tell the world that they try harder.