You don't need a deed poll to change your name. You can just adopt a new name, simply by using it, and call yourself what you want. A marriage certificate can act as proof of a name change, and that applies to both women and men. So, a man can adopt his wife's family name if he wants, and a marriage certificate is documentary proof of his new name. It actually says this on the link. https://deedpolloffice.com/advice/man-getting-married I'm still considering adopting my wife's name at some point. It's far cooler than mine!
But in order to use her name when you need to change your name on your passport, you are better off to have DeedPoll done first to avoid any fuss from Home Office! No need for solicitor to stamp it!
As it says on the link, men and women are treated the same when it comes to name change through marriage. As a man, you can renew the passport into your married name by using a marriage certificate. How many women get a deed poll when adapting to their husband's name? It's the exact same principle. Then, once you have the passport, you have the definitive ID. A deed poll acts as an extra proof, but is not necessary if you are changing to your spouse's surname.