It’s absolutely part of your role as a barista.
Don’t think too much about the customer connection, but view it more like this:
Did you make the customer feel welcomed, we’re you friendly and inviting (to the third space), was the order correct? Was the order taking a super long time? The customer connection score is essentially how inviting, welcoming, and correct you’re getting orders.
It can also account for quality of drinks (like if coffee is sitting too long between brews or if refreshers aren’t being shaken).
It sucks that baristas aren’t as educated in the background information as say SMs or shifts, but to be fair, shifts also have somewhat limited information to connection scores.

Customer connection is essentially viewed as your customer service score (are you satisfying your customers with the service you’re providing). Ie are drinks made right or are they poor quality, are barista kind and compassionate and not bitchy or rude, there’s a lot that goes into customer connection scores (especially as you think outside the box) such as the likelihood that a customer will come back or transition into a regular…IE high quality products means repeat service.

My personal opinion is, if there’s no “connection”, to be made, don’t make it. i can tell when a barista is forcing a conversation or a connection and it makes it feel like their part of a script and they don’t actually care. i’ll be extremely friendly, but i’ll only ask them what their doing tonight for fun or just make some kind of random comment if they seem like their up for discussion or seem to be having a bad day. if it’s an average joe or a person blatantly in a rush or if i genuinely don’t have time for a connection, i’ll just make up for it by smiling more or being more charismatic. i talk to customers like nice people coming through a drive thru that are living normal hard working lives, not like the average Starbucks Partner to Customer connection. i brought this up to my manager and she is actually proud of me for making my connections genuine, and she agrees. connections also come in ways like if a regular customer comes and you ask them if their getting the regular, that’s also technically a connection. if customers are coming back and appear to be happy when they leave, then your doing fine.

Yes it is a part of your role. Here's more background: customers who use the mobile app will periodically get a short survey after their visit. The survey is one question and the intent of the question is something to the effect of, "Did Starbucks go above and beyond to get to know you?" these survey answers get counted and are what make a store's customer connection score. Stores are expected to make customer connections to keep that score up. Starbucks wants to be more than a coffee company, they also want to create a genuinely inviting and caring culture; this is how they do that and why its so important to them.

I can't say why your old store didn't seem to focus on it, but these pushes come in waves or may be pushed when your store's connection score takes a dip for whatever reason.

I understand you might have some nervousness about it, every green bean goes through that at some level. I encourage you to push through it because honestly it's one of the more rewarding if not the most rewarding aspect of the job. It can easily brighten your day to see a customer you've put effort into getting to know.

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