https://en.wikipedia.org/wiki/Clandestine_Marriages_Act_1753
https://www.citizensadvice.org.uk/family/living-together-mar...
https://www.familylives.org.uk/advice/your-family/relationsh...
Edit: also a related term, parental responsibility...
https://www.bma.org.uk/media/1840/bma-parental-responsibilit...
Edit2: also you can assign each other as "Next of Kin".
https://www.royalfree.nhs.uk/patients-visitors/advice-and-su...
Edit3: also note, married couples don't automatically have "Lasting power of attorney" which grants partners a say in their medical care, and being "Next of Kin" is not an equivalent.
https://www.ageuk.org.uk/information-advice/money-legal/lega...
So... don't get married, sign your name on the birth certificate, and fill in that NHS card