I have, on many occasions, received references/letters of recommendation from academics with their student copied in.
Some do it as a matter of course for transparency so that students know what's being said about them or to confirm that they have actually done it (profs are notoriously bad at getting refs in on time!). And some do it in error.
In the cases I have experienced, this hasn't been an issue at all. What is at stake, at least immediately, is the quality of the reference, not whether or not the student knows its content.
Unless there are some very stringent rules in place at the receiving institution, I think that this is a non-issue.
It would also be worth bearing in mind that, depending upon your jurisdiction, the content of references (ie personal data) may actually be admissible upon request. See the University of Reading guidance on this in the UK. So references aren't necessarily private anyway.
The only real issue I can foresee is if the content of the reference isn't great and your place to be declined. Although this one is up to you how you respond.
In answer to your question, if you are really concerned, I would suggest you send your professor an email to thank them for the reference and casually/politely to ask whether it is normal practice to cc the student in to these kinds of letters. The prof will then hopefully give you their view.
I would advise against contacting the receiving institution as you might risk sounding a bit silly/panicky (sorry!).