Drawbacks of making public which journal a manuscript is submitted to, while it is under review?
I think a main reason why many find it preferable to not list the journal name is because it can come across as presumptuous and doesn't really advantage one over not listing the journal name, anyway. For an established researcher (such as, probably, your advisor), the few papers in submission probably do not bear too much influence on his/her reputation, compared to an early-stage researcher for whom these comprise a large percentage of the publication record, so there isn't much to gain.

Two alternatives that may allow you to list your submitted publications in a manner that is consistent with your advisor's are:


Do not mention the journal, but mention that the paper has been submitted. This shows most of the important details of the paper and allows you to record it on your webpage without explicitly naming the journal. E.g., 


  Student A and Doe J. (2016) "Our awesome paper." Under review. 12 pages.

Depending on your field and the journal to which you submitted, you could post an archival version online (say to arXiv). Then you can reference this on your webpage until it is accepted in a journal.