The claim is false.
Specifically, this official reference says of safety razor blades:
Goods are subject to VAT standard rate.
VAT rates are set out here.
I expect the 'grain of truth' is that some aspects of health are VAT exempt (see boxes marked health and healthcare above), for instance dispensed prescriptions and incontinence pads. I expect the logic is that neither sanitary products nor razor blades are to treat a medical condition. Neither, for instance, is toothpaste.
On the other hand, as per the above link, VAT of on sanitary products is at a reduced rate of 5%. This reduced rate of VAT on sanitary products was due to a campaign in culminating 2001 (poor reference). Prior to this VAT on sanitary products was at the normal rate (which from memory was then 17.5% but is now 20%). As far as I can tell the VAT rate on razor blades did not change (save for changes in the general rate of VAT); therefore the claim is unlikely to have been true in the past.
The provisions of what may and may not be at reduced rate VAT are set by the EC, specifically within Article 98 and Annex III of the principal EC VAT directive (2006/112/EC), the relevant section of which says:
pharmaceutical products of a kind normally used for health care, prevention of illnesses and as treatment for medical and veterinary purposes, including products used for contraception and sanitary protection.