Have a look at the anti-spam protection of the affected forms. If there is none, you have been lucky so far. If there is one, it has probably been overcome and you will have to improve it. Unfortunately, there is no 100% protection against this. Without any information on the form technology used, it is also not possible to give you any more precise information.
The reason why you receive so many in such a short time is that such bots sometimes appear in waves – especially when they realise that they have overcome a form.
Which forms are you referring to here?
Many of them signing up to news-emails and also opening accounts on webshops (lots of wordpress webshops).
If you don’t have any forms on your website and the emails aren’t coming through your WordPress-based website at all, then it’s related to something outside of WordPress’ control. Unfortunately, we can’t help you with that here. You could contact your email host’s support team to clarify this.