For housing, you will need to provide some proof of disability, for blue badge, ditto. Each service where you get support, extra points, or whatever, will have their own burden of proof that is required (sometimes consultant's reports, or evidence of being in receipt of certain other benefits that have had their own burden of proof requirements in the application process).
The requirement to be registered as disabled in order to x, y or z does not exist, because there is no register to be registered on. Whether or not you want to have your disability recognized for share to buy, there is no disability register. You can 'prove' your disability in any number of ways, (including having already jumped through the bureaucratic hoops to have a blue badge, or be in receipt of DLA - both of which apply for us) but being on a disability register ain't one of them.
Dd2 has a blue badge. She is in receipt of HR DLA with mobility element. She is disabled.
She is not registered disabled. There is no disability register.
You will need to check the small print of the share to buy scheme to identify which burden of proof they require to accept your application as a person with a disability.
But there is nowhere you can register as disabled that will provide you with the burden of proof required.