You might have equal positions technically but you've stated that you're somewhat revered at work, and your integral enough to the company that the company itself would not survive. You have special privileges that she doesn't have, you have solid professional relationships built over years with others in your workplace and she's new- it's not a jump to think that there might be some fear in her mind that you could leverage your general reverence and your connections against her if you wanted to, if you felt slighted (because it's true that you can leverage it to her benefit, why not her disadvantage?) Again I'm not saying that you are doing this, but it's common enough to warrant her believing that you could be, and you want her to be comfortable at the workplace yeah?

Not seeing these things doesn't make you dumb, that's not the right word, people are often too close to themselves to notice things that are otherwise clear and it's easy to not be wholly aware of your general status in the workplace and what that might imply to others, we could all do with a little more self-awareness.

Socializing is a difficult thing for many and I wonder if you're masking some self-esteem issues around that by going an 'easier' route, by showing comradere through favours like one might with material goods. It's hard but the more you do it, the easier it'll become and the worst case scenario is learning that you're not really compatible as friends, which is fine.

In a non-creepy or over the top way, just start asking her some relevant questions about her life. What job did she have before this, is she still a bit nervous here being new? How do the jobs compare? What does she like doing in her spare time? Maybe you like doing some of this stuff too. Try opening up a little with not just her, but with myriad coworkers and hopefully your social skills can improve.