Can “partial to” and “vulnerable to” mean the same thing?

"Partial to" means "In favor of." "Vulnerable to" means "At risk from." They have different meanings.

No. Being partial to something means you like it, You have a preference or fondness for it.

No partial to is like saying "likes". I am partial to cream in my coffee.

As others said, partial to means likes. But in your specific example that could give a similar vibe. If he likes the lies she's telling him, he's not going to doubt them.

No, "partial to" means you have a liking for something--"I'm kind of partial to chocolate ice cream, so I'll have that," for example. In your example, I'd just use "vulnerable to." It's perfectly fine.

In this one particular case, maybe, but not in general. I think "vulnerable to" works better here.

No, they can’t.

Absolutely not.

Not really. In this specific example you gave, they are interchangeable, but only because to be partial to a lie is to believe it (at least one some level) because to be partial to something is to like that thing. If you like a lie, that does make you susceptible to it.

Partial to means you like something. Two different meanings.

No. They can sometimes fit in the exact same place in a sentence, but they convey slightly different meanings. “He was partial to a pretty face” means he likes girls with pretty faces. “He was vulnerable to a pretty face” means girls with pretty faces can take advantage of him. In this context you’re not really vulnerable to something you dislike, so “vulnerable” implies partial. But vulnerable has extra layers of meaning. “His insecurities made him partial to her lies” makes sense, but doesn’t mean the same as vulnerable. You may wish to look up “susceptible”, which basically means “vulnerable to a particular item or strategy”.

If you are looking for a different word than vulnerable, you could say "his insecurities made him susceptible to her lies." The nuance is perhaps slightly different, but not too different.

It's kind of like you're choosing between, "I love how he treats me," and, "I hate how he treats me," and then asking us, "does that mean that 'love' and 'hate' mean the same thing?"

Not generally. Being partial to something means being favorably disposed toward or preferring that thing.