Firstly, there are many branches of maths which don't currently have their own tags: analysis, calculus, algebra, trigonometry, topology, and so on. You could make a case for creating tags for all of these, but that would be a lot of tags to create that currently have no puzzles assigned to them. You also say "puzzling isn't meant for Algebra", but I bet you I could write an interesting puzzle on every one of the tagless topics I've mentioned!
Secondly, your argument could be extended to say that any tag for a topic whose subtopics have their own tags should be burninated. So [probability], say, should be abolished in favour of [random-walk], [gambling], [dice], [cards], [coin-tossing], [random-selection], and whatever other subcategories we can find? It can be useful to have a more general tag even if its subtopics are covered by other tags.
To take an example from Science Fiction and Fantasy, a more well-established Stack Exchange, there is a Harry Potter tag (the second most popular tag overall) which covers any question relevant to the Harry Potter universe, but there are also tags for Voldemort, Horcruxes, Severus Snape, Hogwarts, J K Rowling, and so on. The tags for Tolkien, Lord of the Rings, The Hobbit, The Silmarillion and Middle-Earth are all among the most popular tags there, but any question tagged Tolkien or Middle-Earth is also likely to be tagged with one of Lord of the Rings, The Hobbit, or The Silmarillion.
To take an even more well-established example, look at the tags on Meta sites such as this very one we're on. Every question must be tagged [bug], [discussion], [support], or [feature-request]. Now let's say you want to report a bug related to one of the site's features. On a broad level, the site doesn't have many features: posting questions, posting answers, posting comments, voting, starting bounties, tagging, chat. Each of these has its own tag. Does that mean the [bug] tag should be abolished? No! General tags have their uses.
Thirdly, as squeamish ossifrage mentioned, there are some puzzles which are mathematical in nature but a) are related to maths as a whole (I couldn't find an example of this offhand) or b) are such that the OP doesn't want to specify what kind of maths is involved for fear of making the puzzle too easy (such as this puzzle of mine).
Finally, let's recall what tags are for. As stated in the help centre, tags are useful for helping people to identify the questions that are most interesting to them. There must be many people (such as myself) who are interested in a wide range of different mathematical puzzles. It's much easier to search through all puzzles tagged [math] than to have to search under many different tags for all the different parts of maths; and if people decide to focus on just one or two of their favourite areas so as not to have to search so many tags, we might lose a lot of high-quality contribution in the tags they're no longer following!