As Jen explained, that particular federal statute (Access to Information Act) only applies to the federal government.
Most access to information and privacy (ATIP) rights are provided at the provincial level.
In british-columbia, the main relevant legislation is the Freedom of Information and Protection of Privacy Act. Every university in British Columbia is a public body subject to the Act (Schedule 1, "educational body").
Notably, the law provides exemption to disclosure when it "would be an unreasonable invasion of a third party's personal privacy" [s. 22(1)]. The disclosure is presumed to be unreasonably invasive when
22(3)(h) the disclosure would reveal
(i) the identity of a third party who supplied, in confidence, a personal recommendation or evaluation, character reference or personnel evaluation, or
(ii) the content of a personal recommendation or evaluation, character reference or personnel evaluation supplied, in confidence, by a third party, if the applicant could reasonably be expected to know the identity of the third party
Normally, when refusing disclosure of information supplied in confidence under the "unreasonable invasion" section, the public body must nonetheless disclose a summary of the information. However, an exemption further applies:
22(5) On refusing, under this section, to disclose personal information supplied in confidence about an applicant, the head of the public body must give the applicant a summary of the information unless
(a) the summary cannot be prepared without disclosing the identity of a third party who supplied the personal information, or
(b) with respect to subsection (3) (h), either paragraph (a) of this subsection applies or the applicant could reasonably be expected to know the identity of the third party who supplied the personal recommendation or evaluation, character reference or personnel evaluation.
If there is only one reference chosen by the applicant, 22(5)(b) would certainly apply. If there are two, it is very likely that the exemption applies. But if there are three, four, etc.? The exemption may still apply, but it would depend on the circumstances to judge whether the applicant could reasonably be expected to know the identity of the supplier of reference. For example, a summary containing the average of numerical scores supplied by several references would probably be treated differently than a summary that would contain evaluations regarding a specific role the applicant held.
In Langara College (Re), 2016 BCIPC 1 (CanLII), the Commissioner found that s. 22(5) does not prevent the disclosure of peer grades (without disclosing the identities of peer graders) received by a student in a class of 36 students.
Note that provincial laws differ. In Alberta and Quebec for example, letters of reference supplied for admission purposes usually have to be disclosed by public bodies.
And if you are an employee of a public institution and provide letters of recommendation as part of your official duties or employment, additional access to information legislation may apply on your own side.