http://www.healthinfoprivacybc.ca/confidentiality/when-can-and-cant-they-tell-others is a pretty good summary. Different rules apply to private practices than public clinics and hospitals. I will assume that the clinic on campus is private. This is a summary of the summary about who your information can be shared with:
- Health care professionals can share information within your "circle of care". Specifically, they are allowed to assume your consent to this but you can explicitly withdraw that consent. This would include doctors within the same practice.
- Admin staff can access your information for administrative purposes.
- Anyone you have authorised them to share it with e.g. relatives, friends etc.
- The Medical Services Plan for billing and admin
- If you are unable to drive
- If there is suspected of child abuse
- If you are wounded by a gun or a knife
- If you are a danger to others
For your specific questions:
I asked about it and they said it's confidential, but confidential to the clinic.
Correct, unless you explicitly revoke this.
the counselling department can share information with the doctors
This is tricker, these people may be either within your "circle of care" or they may be part of the same organisation. Notwithstanding, councillors are not doctors and are governed by the everyday laws related to confidentiality i.e. information given in confidence is confidential and everything else isn't. If you are told the limits of the confidentiality i.e. they tell the doctor, then those are the limits unless you renegotiate them.
he would know I only have one kidney?
Well you said "the counselling department can share information with the doctors" and this would require the information going the other way i.e. the doctor sharing with the councillor. Even if this type of sharing was OK in general (and I'm not sure it is, see above); the information shared should only be what is required for the councillor to do their job - the number of kidneys you have is probably irrelevant to this.
What laws apply to situations like this where confidential information in one entity (medical office) decides to share it without the consent of the patients to another entity (the counselling dept.)?
Well, we are not sure there are 2 entities: legally there may only be 1 - the university. Anyway, the laws are the Personal Information Protection Act and common law (Smith v. Jones,  1 SCR 455)