What you are being asked to provide is in effect grant of probate.
If someone dies having left a will the executors apply for grant of probate to deal with any assets and property.
If someone dies intestate with no will, then you apply to the probate registry for Letters of Administration which is in effect the same thing but done through a slightly different process because there is no will.
If there is no land/house involved there is actually no need for probate/letters of administration and it will usually be dealt with informally under the smaller estate rules.
In cases like that, the institution will normally allow the personal representatives (executors by another name) to deal with the estate by simply signing a statutory declaration wherein they declare that they are entitled to deal with the estate. Certainly, banks and building societies will do that.
The difficulty is that there is no statute that compels them to do that and in the case of shares, whatever the value, they would insist on grant of probate or letters of administration. Ask the insurance company if they will allow you to deal with this by way of statutory declaration. If they will, they will have a standard form. If the insurance company insists on the grant/letters of administration you have no alternative but to apply for probate. You don’t necessarily have to be a relative or next-of-kin.
Here is a link to the process https://www.gov.uk/wills-probate-inheritance/if-the-person-didnt-leave-a-will
Can I clarify anything for you?