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Tank
Patience is key here when countering Brigitte as a tank.
Brigitte has a ton of displacement and a sneakily high amount of damage output. Knowing this picking heroes like Winston and Wrecking Ball into her are less than ideal.
Sigma has a strange love-hate relationship with Brigitte as well. Not only do they complement each other, but she can make playing him tricky.
It's not that you shouldn't pick Sigma, it's more so that you have to be very purposeful with your Accretion. Wasting it on a Brigitte shield is low value. However, so long as Sigma can poke from a range, you can swat her away fairly easily.
When learning to counter Brigitte, the key is about knowing when to commit. Let's say you're posturing your team for an fight and the Brigitte is just swinging away at you. This preemptively allows her to begin to heal any chip damage.
Knowing when to push forward and step into her space is the key to countering her as a tank.
Tank counters for Brigitte include Ramattra, Reinhardt, and D.Va.
Ramattra functions much like how we described Sigma should play.
Reinhardt already wants to be up close, so being able to punish and instantly cause Brigitte to expend resources early is valuable.
And D.Va has a great time handling her displacement with how low of a cooldown her boosters are.
Support
Ana is rarely bad when trying to shut down an enemy support. Blanketing her team with Biotic Grenade is an easy way to always put your team ahead.
However, against an enemy Brigitte, it's that much harder for them to recover those precious moments where they couldn't be healed, granting you a life lead.
Kiriko does a really nice job of not being harassed as much as other supports by Brigitte.
She can shift through walls and reposition quite quickly, her healing isn't affected by any type of displacement. Kiriko is a more safe pick into Brigitte.
Heroes that are highly positional like Lucio and Zenyatta are going to be picks to attempt to avoid.
Detaching Lucio from a wall with a well-placed Whip Shot might as well be a death sentence and Zenyatta really needs to keep a strong line of sight on the entire fight. Being able to Whip Shot him into a corner or off high ground is really costly.
Want to ignore her completely outside of Rally? Moira is a super safe option if your team is in need of a throughput healer. Moira's pure healing alone might just win a healing-based war of attrition.