Try this whois [1] and see if it was updated to reflect your current registrar. Also look on that page after your query for "Registry Server URL" that may have more data. If that does not help, try making a change to the domain on your new registrar and see if it gets accepted or rejected. following change control process in your company of course
If you literally just transferred your domain there can be a small delay in the whois updating. Don't panic, it's mostly harmless. These are old and often overwhelmed systems that update via cron jobs and in some cases they use emails. to stop50's point
Also log into namecheap and see if they still list your domain in the domain management part of the site. Log into your new registrar and do the same. The domain should only be in one place and ideally with the "locked" status on your new registrar. If the domain is only manageable under the new registrar then you should be fine and should be able to ignore the email. If still in doubt get on the phone with your new registrar and they can validate if it moved successfully.
If I recall correctly, namecheap is just a reseller from enom/tucows. See if you can contact eNom/tucows [1] and explain the situation. Perhaps namecheap is trying to get one last payment out of you? Something sounds off about all this. Maybe conference in namecheap and enom together after talking to enom or have them conference in their people. They probably have contact information that is not publicly available.
If that path is not useful then perhaps pay for the domain so you don't lose it, then have your legal team send them scary letters or take them to court. I am not a lawyer but I have handed off things like this to a companies legal team to sort out for other "registrars".
