Personally, I tend to think of Matlab as an interactive matrix calculator and plotting tool with a few scripting capabilities, rather than as a full-fledged programming language like Python or C. The reason for its success is that matrix stuff and plotting work out of the box, and you can do a few very specific things in it with virtually no actual programming knowledge. The language is, as you point out, extremely frustrating to use for more general-purpose tasks, such as even the simplest string processing. Its syntax is quirky, and it wasn't created with the abstractions necessary for projects of more than 100 lines or so in mind.
I think the reason why people try to use Matlab as a serious programming language is that most engineers (there are exceptions; my degree is in biomedical engineering and I like programming) are horrible programmers and hate to program. They're taught Matlab in college mostly for the matrix math, and they learn some rudimentary programming as part of learning Matlab, and just assume that Matlab is good enough. I can't think of anyone I know who knows any language besides Matlab, but still uses Matlab for anything other than a few pure number crunching applications.