I have a similar issue with Outlook on the web browser (Chrome) but it happens only with the attached PDF file, once click on download, the file is downloaded to the "Downloads" folder and automatically opens with the local PDF viewer. It bothers me to close the local PDF viewer and move the file from the Downloads folder to the proper folder. instead of choosing the right folder in one fly.
I looked in the Outlook setting but could not find any option related. I struggled with this issue for months until today I found out how to fix it.
In my case, it has nothing to do with Outlook but the browser setting. I went to the Chrome: Settings/Downloads
Made sure the "Ask where to save each file before downloading" was checked. (This option was always been checked since before I had this issue.)
Then I pushed the Clear button on "Open certain file type automatically after download". Viola! It's done. **You will never see this option if you never accept some options in the browser like what I mentioned below.**
I found that this happened because I had unintentionally accepted the browser pop-up menu "Always open with system viewer" for the PDF file type.
** It will automatically download the associated file type to the Downloads folder and open it with the system viewer even if the "Ask where to save each file before downloading" was checked. **
Hope this will help someone who has the same issue with Outlook on the web browser.