I certainly hope that you haven't broken the law by driving without a license. Most of Canadian refugees do not have the funds to buy a car and rely on public transportation. Also most refugees don't arrive with a US driver's license or good English/French language skills so need to complete graduated licensing to learn the rules of the road in North America.
The reason why Alberta does not issue DL to refugee claimants is the province thinks there are identity verification issues with the Refugee Claimant Documents that the Federal Government has been issuing. This is the exact wording:
Refugee Claimants
If you are in Alberta under a pending or denied refugee protection claim , you do not qualify for GDL E xemption, nor are you eligible for an Alberta driver’s licence or I D c ard, re gardless of whether you have a W ork P ermit issued by the f ederal government.
It has been verified by federal authorities that government identity documents (Refugee Claimant Form and temporary foreign Work P ermit) may be provided to refugee claimants without ve rifica tion of their identity. These W ork P ermits are most often issued in conjunction with a claim in order to enable you to obtain employment, thus relieving you of the necessity for social assistance while you await the resolution of your claim. The Wo rk P ermit does not co nfer official reside ncy status upon you , therefore you are ineligible to obtain any Alberta identity document ( such as an Operator’s licence or ID Card ) until you receive official status within Canada. The W ork P ermit specifically sta tes “DOES NOT CONFER TEMPORARY RESIDENT STATUS” under the remarks section.
https://www.alberta.ca/assets/documents/drivers-licence-gdl-exemption.pdf