I've had an issue with the same flavour as this but a bit different in that my DH spends regular time one to one with a younger woman for their shared hobby. It's been going on for 10 years and I have spent the last 10 years trying to quash my insecurity, tell myself to stop being jealous/possessive etc, while DH has essentially kept the view the whole time that it doesn't matter because he's not and wouldn't cheat. But infidelity isn't just physical. We've met up all together too and that hasn't changed my discomfort. And as a friendship develops it becomes more and more natural or normal to be more emotionally intimate and in my experience I've felt like a boiling frog, never realising it was slowly getting too much for me in my haste to be cool, not be controlling etc.
Anyway, the last few months I've finally had the realisation that it doesn't matter what other people say they'd be comfortable with, ultimately if you aren't feeling good about a certain interaction or relationship THAT'S OK and your partner should take it seriously, not minimise you and ultimately should want to prioritise your feelings and your relationship. Obviously if you feel discomfort about every single relationship or friendship your partner has then maybe there's an inner reason that's you. But if generally that's not a problem for you but one person is bringing those feelings up, listen to your gut.
Please grab 'not just friends' by Shirley glass - get amazon Kindle on your phone and download it today. It's largely about recovering from affairs which won't be relevant but the whole beginning section is about how emotional intimacy and friendship can be a precursor, signs that it's crossing boundaries rather than just a friendship etc. It really helped me finally see that it's OK to have boundaries of what feels comfortable to you and that actually within a marriage there should be some lines - walls and windows is how she frames it. It's also helped me see it's OK to have an expectation that your emotional wellbeing should be important to your partner. I thought my negative feelings were mine to get rid of and inherently unfair on him.
Don't be me and let everything go because he hasn't 'technically' done anything wrong. I'm now here 10 years in and he's just the other day conceded that maybe he could no longer do the hobby with her but has also revealed he sees this as possibly temporary until I'm in a better place. I'm constantly second guessing myself and it's all because he wasn't prepared to make a concession to me for the sake of our marriage. It really sucks.
Definitely go along for the drink it's not weird. It's a good sign she is trying to include you as it suggests she has some awareness it could be difficult for you. But why do they need to message almost daily? Does he message with that frequency any of his other friends? What are they messaging about?