They probably are aware. Anxious people are often self-aware their thoughts aren't rational, but they still have them and will still reach out to get external affirmation that they're not as bad as their head is making them out to be. Pointing out how their behaviour is making you feel can help snap them out of that, i.e. its making you feel like they're not listening to you or trusting what you say, making you feel like you're talking in circles, etc; pointing out that they're "self-obsessed" isn't going to help anything. Either they're anxious and they now feel like they're just as bad as their thoughts are making them out to be and feel even worse/shut down and avoid speaking to you whatsoever, or they're anxious but get defensive because it feels like an attack and now you're in an argument that goes nowhere.

They might already be working on it and asking for affirmations is something their therapist suggested to try and get them out of their own head but they're taking it too far, they may be in therapy but still working on their anxiety and not yet at a point where they're good at catching these things. Unless you know for a fact what they're doing outside of your interactions with them, you probably don't know if that's something they know and are working on.

The tip's centered on things they can change in the moment, aka lowering their voice, etc, not who they are as a person. Its like how you don't point out a physical issue or change on someone if they can't fix it in the moment- having something in their teeth yes, gaining a few pounds since the last you saw them absolutely no. If they're asking if they're annoying, point out what they can fix in the moment and focus on how its making you feel, not what you think they feel/think that's causing the behaviour.