I noticed that clicking the "Download" option/button in Google Drive, Slack, etc. doesn't prompt the user for where they want to save, and with what filename. It just goes to my "Downloads" folder with a default name. Is that standard?
Yes, that's standard. The filename will be whatever the host site named it.
There may be situations that call for a different experience (as another answer suggested) but if the question is whether or not it's standard practice to use the given filename at download then I would say the answer is yes.
The only possible exception I can think of is downloading a PDF from an online PDF viewer. You may get a full save dialog box in some cases.
I would make the choice based on user expectations. Most browsers default to downloading content to the "Downloads" folder hence you can be fairly confident that users will be familiar with this construct and will know where to look to find their files. With other applications e.g. a business app with sensitive information, this may not be the best choice so you would likely want to prompt the user for the download destination