The Atari 2600 had a screen saver feature, where the colors of objects in the screen were randomly changed.
Was it the first home console to have such feature, if not who holds the title?
By definition of "Console" to mean a system dedicated to play games or entertainment software, it depends on the criteria. In basic format - yes.
From Wikipedia - Screensaver, History: (emphasis, mine)
The Atari 400 and 800's screens would also go through random screensaver-like color changes if they were left inactive for too long. The user had no control over this. These computers, released in 1979, are technically earlier "screen savers." And prior to these computers, the 1977 Atari VCS/2600 gaming console included color cycling in games like Combat or Breakout, in order to prevent burn-in of game images to 1970s-era televisions
Though the "Brown Box" is considered to be the very first home console in 1967, it did not reportedly feature a screensaver and screensavers were primarily used on early computers with CRT monitors. As written in the snippet above, the Atari 2600 had a screensaver but it should be noted that only if the ROM included it - that is, "screensaving" was contingent on the medium and the programming of the medium.
As far as my own research, consoles required a cartridge or disc inserted to boot and function or had the ROM(s) permanently attached to the internal PCB until the Sega Saturn.
Many other consoles including the Nintendo Entertainment System to the 3DO Real Interactive Player and Amiga CD 32 did feature animations asking for players to insert a medium, which can also be argued as a screensaver on a functional level but served more as a mechanic to prompt the user to act.
This means Sega Saturn was the first home console that would boot without a medium inserted(*)(**) from and allow the player to use other preinstalled device features (such as playing media and manipulating storage devices).
Incidentally, playing music on the Sega Saturn featured a flying spaceship as a form of screensaver, which is the truest form of screensaver that was built into a mechanic of the console rather than of the medium.
Therefore it can be argued that while the Atari 2600 was the first console to use a screensaver as a feature of select media, Sega Saturn was the first to deploy it as a standard function of a feature.
There is a YouTube video showing the start-up screens of every major and most minor consoles from 1985-2013 to see how some consoles had iterations of what one may consider a screensaver until the user took action. It's run-time is just over 14 minutes.
(*) Sega CD was an accessory to the Sega Genesis released in 1991, not a separate console developed to operate independent of something else therefore not relevant in context to the question or scope of answer.
(**) Sony PlayStation was released in Japan on December 3, 1994 - 11 days after the Sega Saturn in their country of origin.