The original screenshot shows “banned temporarily” and “number of requests you have made to this website has exceeded the rate limit”.
It doesn’t say “buyer requests”. It could be page requests (like you say, you opened too many tabs).
Though I would have expected it to just block the IP address if there were too many page requests but maybe not. Opening tabs too quickly can sometimes get the captcha thing.
If you haven’t posted a buyer request (in the buyer section) for a long time maybe what CS said is a mistake like suggested. You could contact them and say to them you haven’t posted one in a long time. You could also show them that screenshot of your “manage requests” page showing none.
If it’s too many page requests in a certain amount of time, could it be an auto-refresher was being used to appear always online or to get the latest buyer requests in the seller->buyer requests page?
But since the account shows okay on the main site, and assuming you can currently access it, maybe it was just a temporary thing (related to page requests?) and maybe Fiverr CS made a mistake in thinking it was buyer requests.