Most non-trivial text-rendering systems need to know the language of the text in order to properly render it. This is not only an issue for Japanese/Chinese but also e.g. English/German with different hyphenation rules[0], different rules for ligatures[1] and different rules/preferences for spacing after punctuation[2]. None of these can sensibly be solved by Unicode.
[0] ‘backen’ (to bake) becomes bak-ken when hyphenated at the end of a line, ‘airlock’ can’t be hyphenated at all between c and k.
[1] ‘Schiff’ (ship) should be rendered with a ligature ff, ‘Dampfflut’ (~ steam flow?) shouldn’t. ‘afferent’ probably should have it.
[2] Single space after ‘.’ and ‘,’ in German, whereas French tends to prefer more space after ‘.’.
Chinese, Japanese, and Korean writers face similar issues. The characters they use to write the name of China or Japan, the ten digits, the characters for year, month, and day in dates, and so many thousands of others in Chinese characters are what they all consider to be the same characters. That is not all characters, but it includes so many that insisting on different code points by language would make a real mess. Hong Kong has many characters that are unique to HK Cantonese. So, should Cantonese have a full set of all Chinese characters that are the "Cantonese characters"? How about Shanghainese, then? Or Hakka? Teochew (Chaozhou) or a dozen Chinese languages? Full, independent sets of all Chinese characters for each? Suppose you accidentally used an input method in HK and wrote the name of some Beijing gov't ministry using characters that looked identical to their Mandarin counterparts but were entirely different codepoints? Now what? You can't find your search term? You mess up the database and have two identical-looking keys?
No, Han unification is not conceptually different from unifying ABCs used by English and Spanish speakers, Cyrillic used by Russians or Serbs, etc., except that there are many more characters, so the boundary between what should be unified and what shouldn't contains more items in the gray zone to cause debate. Having no Han unification at all wouldn't solve all problems, it would create all sorts of absurdity.
Also, comparisons like "different ABCs for English and Spanish" are spurious and unhelpful. If you could tell an English "b" from a Spanish one by looking at it, the comparison would be sound.
It's difficult to come up with a logical explanation for why European languages that use their own alphabet get their own codepoints, but ideographic languages need to be "unified", even though the actual letters as used in those languages look different.
The "Han unification" was fundamentally a bad idea, and persists for historical reasons. Back when (some) people thought a fixed-width 16-bit character representation would be "enough", it made sense to try to reduce the number of "redundant" code points. Now that Unicode has expanded to a much-larger code space, I would think they'd choose differently.
Unfortunately, that kind of sweeping change is unlikely any time soon.
Gather 5 linguists, you won't get them to agree on that. Unicode says they're the same, but not everyone agrees with them, and the practical problems are real.
Example 1. Typically, any time you want to mix the 2 languages you're getting in trouble.
Let's say you write a textbook for Chinese people to learn Japanese as a second language. Or a research article in Japanese citing old Chinese literature.
In your text, you'll have to mark specifically which part are in Japanese and which part are in Chinese and use different font for them. If you don't, the characters look wrong.
Example 2. Most phone software don't switch font for languages, so they pick one. Let's say you buy a Japanese phone, it's a Japanese font everywhere. For some reason on a Samsung or Motorola phone bought in Europe, it's a Chinese font everywhere and it looks ugly for Japanese. Wasn't Unicode supposed to be universal, and allow you to read in any language with a device bought anywhere?
On the other hand, I don't think the "I can't write my name" is really a problem for Japanese people, or at least it's not a Unicode problem. Some people have rare characters in their names. They accept and understand not being able to type their name on a computer and having to settle for a close characters. Unicode actually provided more characters than previous Japanese-only encodings like JIS so there are people who could write their name thanks to Unicode.