Government places are claimed by the owners just like business owners need to claim their businesses on Google Maps. But of course, only relevant employees should do so.
You can not create new categories. But you are welcome to suggest new categories in the Idea Exchange. I would not be very optimistic about getting new categories added, though.
As you have received good tips from @MortenCopenhagen I’ve marked his answer as solution, however both of the comments are relevant. The purpose of this feature is to help Local Guides find answers on specific questions or issues when searching on the Connect. You can take a look at: How do I mark comments as solutions.
If you need any further guidance or you face specific issues, you can reach out to any of the Google Moderators.