The law actually contains a list of permitted characters [1]
Your company name can contain curly left apostrophe, curly right apostrophe, and straight apostrophe - but no lower case letters.
There are also a bunch of rules about specific words [2] - so you can't have "Financial Conduct Authority" in your company name without the permission of the government department of the same name.
But that leaves open the door for "FOO[space]BAR" (one space) and "FOO[space][space]BAR" (two spaces) to be registered, so that doesn't really accomplish the goal of "company names must be unique." If case-insensitivity were really their goal, that could easily be accomplished by choosing a case-insensitive collation for their DB.