It’s just a coincidence that the first 13 seasons of Bergeron’s career he was a 50-60 point guy and then he gets matched up on his current line and is much better offensively? His linemates have nothing to do with it. His crap coach that kept leading the team to deep playoff runs held him back.
There are a ton of factors at play. Not to be discounted is that Bergeron has legitimately worked on his offensive game though. His one-timers are night and day to what they used to be earlier in his career. The 2 main factors at play though:
Top-line utilization: It's not so much the ice time he got under Julien vs. Cassidy (which is essentially equal at even-strength from when Bergeron rounded back from his injury ~2010 until Julien got fired vs. Bergeron under Cassidy), it's the quality of ice time. Julien was obsessed with overloading Bergeron with defensive responsibility, defensive starts, hard-matching, etc. Cassidy on the other hand is always putting Bergeron's line out in offensive situations, gives them a ton of O-zone starts, and really doesn't care about line-matching for Bergeron. He's letting Bergeron's line dictate the play, not tasking them with shutting someone else down and letting the offense come off the counter-attack. I think you can apply this to his linemates as well. There's no question that Bergeron is aided with his numbers to some degree by his linemates, but they all do. They all just have great chemistry together. It's not like Bergeron wasn't playing with very talented linemates before when he was putting up ~60 points (Marchand and Seguin).
PP role: I don't remember when exactly they started using Bergeron in the slot position, but I do know that this was something Julien started to utilize. For such a long time, they didn't really know where to use Bergeron, and would frequently try things like putting him on the point. After they put him in the bumper role, he was so good at it that I think you can argue he's the best in the league at that specific role on the PP. He's also getting about a half-minute more on the PP. Julien-coached teams in general just aren't very good at drawing PPs though.