There is a "rule". ( Like FASB statement in the U.S )
The very basic rule is, Commerical Code, Article 11,12.
Article 11 (1) A merchant (excluding companies and foreign companies; hereinafter the same shall apply in this Part) may use his/her surname, full name or any other name as his/her trade name.
Which means even you can even select your name as your company's name.
There are bunch of regulations such as in Article 6,7,8,9 in Company's Act
Article 7 No person who is not a Company may use in its name or trade name any word which makes it likely that the person may be mistaken as a Company.
Granted, here is the ranking of the names of companies that exist in Japan.( The number is how many companies use the word somewhere in company's name )
Please scroll down and pick the numbers to check.
Starting with companies with the name ひまわり、at 25th, ( 427 companies use this word ),さくら at 27th with 414 ( companies ), みらい at 125th with 228, みやび at 196th with 191, みのり at 322th with 159, あすなろ at 340th with 155th, こころ at 360th with 152, あさひ at 367th with 151, ひかり at 421th with 143, たんぽぽ at 467th with 136 companies.
And all the other names are either Katakan or Kanji.
But what we can presume from the data above, is I think, entrepreneurs tends to choose Hiragana as their company's name, when they are related with flowers, lights, souls, future things IMO.
I think the reason why there are so many Katakana companies is because up until late 19th, we did not have European style "companies", I guess. ( And they sound like European, probably. )