I don’t think there is any specific documentation about this. Multiple keys and applications are allowed for legitimate reasons (such as a test and live key etc), but using them to circumvent the rate limit is (based on previous posts by CH staff) frowned on. It has been stated that abusing keys to get around the rate limit can lead to being IP banned without warning. As well as the documented rate limit, there is an unpublished IP limit which I assume is used to police this. I remember them being rather alarmed by someone who had 50 keys or so at some point in the past!
When people ask similar questions they’re often referred to the following:
Acceptable use of Companies House Searchable Databases
The Companies House searchable databases are designed to meet the needs of a wide range of users wishing to interrogate our information on-line.
Due to limitations of equipment and bandwidth, they are not intended to be a source for bulk downloads.
Individuals, companies, IP addresses or blocks of IP addresses who deny or degrade service to other users by generating unusually high numbers of daily database accesses, whether generated manually or in an automated fashion, may be denied access to these services without notice.
I don’t use the streaming API so don’t know about that.