Since Peter answered your first question, I'll attempt to answer your second question.
And, once we've established who conflated the two, what was the biblical rationale for doing so?
Jesus spoke of doing good to others.
Matthew 25:35-40 - ...for I was hungry and you gave me food, I was thirsty and you gave me something to drink, I was a stranger and you welcomed me, I was naked and you gave me clothing, I was sick and you took care of me, I was in prison and you visited me.” Then the righteous will answer him, “Lord, when was it that we saw you hungry and gave you food, or thirsty and gave you something to drink? And when was it that we saw you a stranger and welcomed you, or naked and gave you clothing? And when was it that we saw you sick or in prison and visited you?” And the king will answer them, “Truly I tell you, just as you did it to one of the least of these who are members of my family, you did it to me.”
All Bible quotes in this post come from the New Revised Standard Version of the Bible.
Paul spoke of doing good to others.
Hebrews 13:16 - Do not neglect to do good and to share what you have, for such sacrifices are pleasing to God.
Philippians 2:4 - Let each of you look not to your own interests, but to the interests of others.
There's no question that Jesus and Paul wanted us to help others.
The contemporary question is whether Jesus and Paul wanted individuals and groups (churches) to help others, or have the government help others. Generally, United States conservatives favor individual assistance, while United States liberals favor government assistance.
The scripture is not clear on this question. What we do have is Jesus' words:
Matthew 22:17-21 - Tell us, then, what you think. Is it lawful to pay taxes to the emperor, or not?’ But Jesus, aware of their malice, said, ‘Why are you putting me to the test, you hypocrites? Show me the coin used for the tax.’ And they brought him a denarius. Then he said to them, ‘Whose head is this, and whose title?’ They answered, ‘The emperor’s.’ Then he said to them, ‘Give therefore to the emperor the things that are the emperor’s, and to God the things that are God’s.’
This scripture has been interpreted to support both individual assistance and government assistance. Unfortunately, there is no clear cut answer from the Bible.
And that's why the individual assistance and government assistance debate continues and will continue.