I am making one central assumption which is that you are tax resident in France and my comments are made on that basis.
It is my understanding that there is a full exchange of information between the English and French tax authorities. So any income you receive on investments, savings, bank accounts, sales of assets, is collected by the IR and given to the Impots.
If you are asked to provide your TIN, you should do so, though the choice is yours. However since in our case John Scully and Peter Scawen are known to the Impots and in your case (I assume) you have not supplied your TIN when asked, you will/may show up on the data listing and potentially invite queries.
The fines for non disclosure can be quite heavy.
You are legally required as I am sure you know to declare to the Impots, your bank details and all world wide income. I am quite sure that each country applies the money laundering regulations in a different way so I am not surprised at your experience. However, I would add that I know from personal experience that HSBC are chasing people down so maybe you will be contacted shortly.
It is also my impression that the attitudes towards this issue in France vary from region to region, some people on this blog have reported heavy fines for non compliance, others have reported no action.
So I am not being didactic, just recording my understanding as I do not claim to be an expert, only reporting what I have been advised.
My solid counsel is to get professional advice.