Not mentioning your surname or dad's name is very different to actively avoiding difficult conversation.
At the ANO party, Sage is distracted by avoiding her dad - who she just, y'know, calls Dad, not, "my father honourary Professor Steven Burke" - and doesn't like talking about Tyballs, so why would she want to talk about them when the MC is there to talk to her? To keep the comparison fair, it would be more appropriate if the MC had asked about her family in that conversation and she avoids the matter - the only time that happens is the Ep9 sex scene where the MC struggles: they openly talk about her family (after the dinner of course) except for her surname - that is her being evasive for some reason, but Sage has already established that she's bad at opening up and being emotionally vulnerable (despite being actually pretty good at it) and so the shut down of the conversation is not unexpected or particularly malicious.
Could Sage have talked about her surname more? Yeah, sure, she could have - but I'm, not sure I see why she should have, since she has plenty of baggage around her adopted and birth names. As someone who is adopted, I personally don't talk about my birth family pretty much...ever; my adopted family is my family, and I don't really feel the need to talk about them very much unless someone is asking about them - and even then, I still refer to my mum and dad as...mum and dad, not by their professional titles!
