You're right, it has no legal basis. In fact, if pursued with any zeal, it can even be construed as a con-trick (though I am sure it is not meant as such) to get people to pay up. At the very least it is frustrating and non-productive and does the memory & reputation of the deceased no good.

The best way to deal with such requests is to respond politely that you intend to continue singing those songs and will pay for whatever legal licencing is required to the appropriate licencing authorities. The family of the deceased can then make their claim from such authorities, but not directly from you. Add that you do this in order to be "legal and above board", and with no malice or wish to be obstructive. You should then ignore any further such requests thereafter, leaving it to them to pursue any legal action.

If they become more aggressive after that, I would report them for harassment.

You can also name and shame, of course...