Academic dress (above), Avicenna (below)
Short answer: this is false.
Let's analyze this by looking at the claim.
- In order for this to be true, Avicenna would have to have a distinctive form of dress different from other people of the time. While this image does appear to show him in a two-layered robe and an unusual hat, other images of him show him in a much more standard robe and turban. Even other images of the same statue show that the hat looks much more like a turban. So Avicenna's dress wasn't noticeably different from other people, and his preferred headgear was a turban. Turbans were not used in either academic dress or medieval Christian religious dress.
- It is universally agreed that modern academic dress derives from Western everyday university dress, which in turn is related to clerical dress      . The history of clerical and university garb does back at least to medieval times, probably to around the time of Avicenna. So for the Avicenna theory to be true, it would have to be true that medieval Christian religious dress was derived from the dress of a contemporary Islamic philosopher. This would seem to be extremely unlikely.
- There appears to be no evidence whatsoever for the connection to Avicenna, simply unsupported claims.