Until recently, almost all cold callers (a.k.a. telephone pests, and/or telephone scammers) hid their Caller ID.
However, more and more telephone customers are now choosing to filter their phone calls by displaying the Caller ID, and choosing not to answer anonymous calls. As an example my mobile phone automatically blocks anonymous calls entirely. This has made it more difficult for cold callers.
As a response to this, many cold callers have now resorted to using forged Caller IDs. When you call the stored number, it turns out to be a completely different company from the company supposedly calling you. Or in some cases, the forged Caller ID does actually belong to the company that is allegedly calling you. In either event, if you call the stored number, they will have no record of the call.
The cold callers often choose forged Caller IDs that are never used by their legitimate owners, to make outgoing calls.
So although calls from this caller have a caller ID that corresponds to a legitimate BT number (0800 169 1608) they are almost certainly NOT from BT. This is most likely a phishing call, where the caller will attempt to obtain personal details e.g. bank card/account details. If you are in doubt, ask them for the last 4 digits of your BT account number, associated with your phone number. If they are genuinely from BT, they should have this information.