The regulator told the charity to rename itself after the older education charity Cambridge Muslim College said the similar names were causing confusion.
CMC, which was founded in 2009 and joined the charity register in 2010, complained to the commission that the similarity between its name and that of CIC had led to donations being made to the wrong charity.
The commission issued CIC with a direction to change its name in September, but CIC, founded in 2013 and registered in 2015, has appealed the decision to the charity tribunal.
The case is on hold while the commission carries out an internal review of the decision.
CMC, which in the year to 31 August 2015 had an income of £644,000 and spending of £660,000, has charitable objects of advancing Islam and training Muslim scholars. CIC, which has not yet submitted any financial information to the commission, has the object of providing higher education in Islamic studies.
In a statement to Third Sector, CMC said it had been concerned about the potential for confusion since CIC was established.
"This concern has unfortunately since been proven correct," the statement said. "In the past two years, CMC has been faced with multiple incidents of misdirected communications, including emails, phone calls, letters and, more seriously, misdirected donations."
It said the similarity between the names was a "significant obstacle" for the charities in fulfilling their duty to be transparent with the public about how their donations were used.
The statement said CMC had complained to the commission only "as a last resort" and was very disappointed it had not been able to reach a direct solution with CIC.
"Even before CIC’s registration with the Charity Commission, CMC reached out to CIC to voice our concerns, and has repeatedly tried to engage it formally and informally in finding a solution to this problem in a spirit of cooperation and goodwill," it said.
"Unfortunately, despite these attempts, CIC dismissed our concerns as invalid and did not address the problem meaningfully."
A commission spokesman said: "The commission can direct a charity to change its name if the commission considers it is similar to the name of another registered charity or it is misleading or offensive.
"[CIC] asked the commission to review the direction under the commission's internal decision review service. This review is underway."
He said the appeal had been stayed until the commission completed its decision review.
According to documents released by the charity tribunal, the appeal has been stayed until 30 December and the commission will not be required to respond to it until 27 January.
CIC declined to comment.