Google doesn't have this feature apparently, nor do any of the other search engines I tried, so if someone ever asks you "what can Bing do that Google can't," the answer is RSS feeds for SERPs, apparently!
To subscribe to Bing's SERP, follow the following steps:
1: install an RSS client, which you'll need to use RSS.
2: search for something that you expect to have different results if you searched for it again at a later time, like... uh... I don't know. I really have no idea. I guess the only thing I can think of is if you're searching for something that doesn't exist, and if you keep searching for it, then if one day it does come to existence you'll be notified of it via the search engine? That sounds okay, I guess. Alright, let's say we search for "good social media platforms."
If you do this, the URL on your address bar will contain all sorts of irrelevant parameters. We can remove most of them and leave just:
https://www.bing.com/search?q=good+social+media+platforms3: add &format=rss to the URL, like this:
https://www.bing.com/search?q=good+social+media+platforms&format=rss4: add this URL to your RSS client. You're now subscribed to a search query!
Tip: it's a good idea to set the frequency of updates to weekly or monthly in your RSS client to avoid issues. The results shouldn't significantly change more often than that. The default on some clients is to check hourly, so Bing may think you're a bot if you keep searching for the exact same thing every hour.
News Search
This trick also works with Bing news search.
https://www.bing.com/news/search?q=good+social+media+platforms&format=rssObservations
This trick doesn't work Bing image search, video search, or shopping search.