I don't know all the history because I've only been here a few months, but there is a state of chess.com 2019 video here where Danny mentions analysis on https://www.youtube.com/watch?v=R-Qw5Jsh1OQ what was then the new Android app, so I suspect it was available on the website a while before then.
History of Game Analysis on Chess.com

We're currently in Vers. 3 and if I remember, game analysis using an engine was planned to be rolled out with V3. In the event it was introduced some time afterwards, so that would probably have been around 2018/19 I think.
Why the interest?

We're currently in Vers. 3 and if I remember, game analysis using an engine was planned to be rolled out with V3. In the event it was introduced some time afterwards, so that would probably have been around 2018/19 I think.
Why the interest?
The site is well past v3

Regarding the question, there's this https://www.chess.com/blog/News/meet-the-new-analysis-game-report-retry-mistakes-more
You could also go back through the month in reviews to see what documented changes there have been.
https://www.chess.com/article/view/chess-com-month-in-review-april-2021

We're currently in Vers. 3 and if I remember, game analysis using an engine was planned to be rolled out with V3. In the event it was introduced some time afterwards, so that would probably have been around 2018/19 I think.
Why the interest?
The site is well past v3
Really? The V3 rollout was a major event and was well reported. I remember it being delayed for ages because of teething problems in testing.
If we're past V3 I don't remember reading about it anywhere - what version is the site at now?

The first mention in Beta forum is Feb 1, 2018, see v3 Analysis Report lacks Blunder details.
One major change was from Retry tab to Key Moments.
Changes include coloring Best Move Diff tab and addition of Criticality Tab
The change to CAPS accuracy, soon to be CEE CAPS2 accuracy
Numerous small improvements and settings changes for analysis depth, max depth, engine choice.
Searching Beta forum for: Analysis Report Bug yields 8 pagers of forum topics that will provide a good history.
Does anyone have a timeline of when Chess.com introduced Game Analysis, and any significant improvements they have made to Game Analysis over the years? My understanding is a big change happened in early 2019. Have there been big changes since then? Does anyone know when it was introduced?