Well, pharmacists are not just there to pick medication off a shelf and hand it b out. They also have a professional and medico-legal responsibility to ensure that a drug is safe and at the correct dose. I am a dr- we aren’t infallible and the pharmacists are there as a “second line of defence’ if you like- to make sure dose correct/no concerning interactions with other medicines etc. Some medications can be given at different doses for different conditions, or need to have doses altered in certain situations, so knowing why you are taking it may be relevant to that process.
If a pharmacist dispenses a medication when they should have picked up an error, or should have picked up that it was unsafe, then they will also be held responsible- not just the prescribing Dr (though the majority of the “blame” would be carried by the prescriber). So, yes, they do have a right to ask you about medication agreed between you and your Dr and are within their rights (and duty) to refuse to dispense medication they deem unsafe- or medication where they do not have the information available to them to ensure that it is the case. They would be duty bound to then try and find that information out- e.g. contact the prescriber- but that would obviously delay the dispensing of the medication.
Medication prices are also very variable. It can depend on wholesale price and manufacturer- for instance, when a medication is in low supply wholesalers will often hike the prices up, or sometimes generic cannot be sourced which can lead to price being higher. It really is not straightforward and the pharmacist is probably not being underhand if they are charging different prices at different times.
In any case, yes ask them to change your name. I doubt they are doing it deliberately to annoy you, but if it is important to you, ask them to sort it. Do you care if they internally think you are being a bit dramatic or wasting their time as long as they make the change you ask and do it politely? Overall, it would seem that you might be better to find an alternative pharmacy to take your business to as you are unhappy with the service you have received.