I think people should be kinder and a bit more clear in explanations, the clue is in the OP's username.
OP it sounds like you have two issues:
First you copied a customer into an email chain which was talking about them - this is usually not recommended, a good rule to abide by here, if the customer is included in the email chain from the start, then assume the information is meant for them. If the customer is excluded from the email chain, assume that the information is confidential and should not be shared with the customer. If unsure, ask someone before you copy a customer into an email chain which is discussing them which they are not currently a part of.
Secondly, using the address "To whom it may concern". If I needed to address somebody professionally and I did not know their name or sex, I would use "Dear sir or madam," or "Dear all," for a group of people.
It might help people give advice if you can say what the client/customer relationship is all about - my assumption is that this may be some kind of public service, where the "customer" is the recipient of the service and the "clients" are workers who are helping the customer. However, if this would cause an issue with confidentiality, then don't reveal this on mumsnet.
You say "just joined this year" - the company, or mumsnet?
To go back to the first problem, it seems like this is a problem because usually the expected thing to happen would be that all clients would discuss the case and agree on a course of action for the customer. This would then be summarised by somebody and presented as a course of action with action points to the customer.
Sending them the entire email chain will be confusing, overwhelming, might contain information which is no longer relevant and might contain sensitive information which would be upsetting for the customer to read, which is why it is not usually done. (I am making an educated guess here).
I think your manager is trying to be nice by saying that "To whom it may concern" was a secondary complaint caused by the customer's lady who complained being annoyed by the situation in general. What she means is that you should not be upset by this as a complaint, but it probably is still important.
It is important to use the right address ("Dear Sir or Madam / Dear all", not "To whom it may concern".)
It is more important to ensure that standard communication procedures are followed when sending information to clients and customers.
But both things are helpful feedback to take forward. If you have trouble with things like structuring emails, your manager might be able to find or collate a set of guidelines for you to follow.