Unfortunately that is not possible. You cannot use a logged in users credential to access the ADLS via ADF linked services.
Ideally best way would be to have seperate ADFs per ADLS and give access to the users to the specific ADF only.
Yes, it is possible to get the log details via Activity log to check which user executed the ADF and which ADLS/connection he/she used but you would have to have some custom log set up to capture that