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.