Yes. When a medicine is first invented, the company who invented it can call it what they like. This is usually referred to as brand name. Because they invented it, they have the right to be the only company allowed to make it for certain number of years. This is known as a 'patent'.
When the patent expires, other companies can make the same medicine and call it whatever they want. They usually use the name of the active ingredient and these are known as 'generic' medicines.
For example, Nurofen is a branded medicine and a supermarket's own label ibuprofen is a generic medicine. Both branded and generic medicines in the UK must be made to the same standards, otherwise they cannot be sold to the public. However, generic medicines are usually much cheaper than the branded medicines.
When buying medicines have a look on the packet of branded medicines to see what the active ingredient is. Then find the supermarket's or the retailer's own label which has the same name as the active ingredient. Sometimes, the two medicines won't look the same, for example the colour or shape of the tablet may be different. But if the generic (own label) medicine contains the same active ingredient in the same amounts as the branded medicine, then you can be sure that you are getting the same active medicine.