I grew up in a liberal, cis-focused culture that was not aware of the existence of trans people for the most part. Because of this, it took me way too long to realize that being trans was even an option, and even longer than that to realize I was trans.
Growing up in an environment like that, it's easy to view the idea of being transgender as something that gets tacked onto a default cis lifestyle.
But I still grew up with liberal tendencies, and liberals still have a tendency to offer some vague kind of consideration to marginalized groups, if only for keeping up appearances. And so it was perhaps inevitable that I approached the idea of transgender people with the cultural message of "Well, trans women are actually the same as cis women! No differences, aside from different starting points!"
So I felt a little weird when I started transitioning, and I realized I generally felt more comfortable around trans women than I did around cis women. Did this mean I had internalized the idea that there was something fundamentally different about trans women? Did I not actually view cis and trans women as equals?
So like... I have another example of misguided liberal thought to compare to this. Another message that was ingrained in me when I was growing up, on the topic of racial equality this time, is "black and white people deserve equal rights because there are no differences between black people and white people. Both groups are groups of human beings." And at face value, that seems like a pretty solid message, right?
But I'm white, and there initially wasn't much reason for me to question this ideal. But as I got older and started hearing more about the experiences of PoC, I came to understand that there is still a racist implication in this message, because white people often frame it to mean that black people can be equal if they choose to match the cultural expectations of white people. Because there's still an underlying assumption among many white people that "being human" is equivalent to "being culturally white".
Different cultural groups are going to have different traditions, beliefs, and values. Because of this, it is a falsehood to say that different cultural groups are the same as each other, or should have the same goals. However, it would also be false to say that this makes any given culture lesser or greater than another, because all cultures develop their values in response to the shared premise of trying to figure out how to survive, thrive, and be happy on this godforsaken planet. Different people are taking different approaches from each other, and we should be thankful that this is true, because it means we can all learn new things from each other.
So let's take this idea learned from ethnic/geographical cultures and compare it to gender cultures. My conclusion is no, I don't think trans women and cis women are the same. I think the reason liberals shy away from this statement is because they think the implication of it is that trans women are like men--and I absolutely do not think that. I know trans women are women. I know trans women and cis women are together under the broader umbrella of "women". I just don't think this fact in itself means that trans women are the same as cis women.
Just as people from two different cultures come from two different backgrounds, and develop their own personal views and values in response to that background, trans women and cis women also come from different kinds of backgrounds that fundamentally shapes how they approach ideas of gender and everything else in life. Some part of that is biological, sure, but most of it is cultural, in itself.
I don't think there's anything wrong with this. The more specific parts of your identity you share with another person, the more you're gonna have in common, and the easier it will be for the two of you to communicate with each other (generally speaking). Cis women are generally going to have an easier time connecting with other cis women than with trans women, and trans women are generally going to have an easier time connecting to other trans women than to cis women.
However, this does not stop trans women and cis women from already having a substantial amount in common with each other just by nature of both being women. And it doesn't make being trans better or worse than being cis. Just... different. And trans women and cis women can come together and learn from each other because they have those differences from each other. And again, that's a good thing.
...I've focused a lot on women in this post, in part because I'm one, and in part because, let's be real here, usually when liberals or conservatives think about trans people, they're thinking about trans women specifically. But everything I say here applies to other gender identities, too. (Well, men primarily, since that's the only other element of the liberal assumption binary. But hopefully you get what I mean.)
When you meet another human, you will always be able to communicate and relate to each other on the basis of being human. Whatever more specific things you share will help foster communication further, but don't focus on only that. Accept and respect the other's differences, and focus on the excitement of the fact that it means they may have new thoughts and experiences to offer to you that you wouldn't have otherwise.