Well, I'm from Pennsylvania, founded by Quakers, meaning that the laws when we were a colony and then when we were a U.S. state, were heavily influenced by Quaker practices. For Quakers to marry, they just need to say in front of people that they pledged themselves to each other. For that reason, until quite recently, PA had extremely casual marriage laws, in which that's all people had to do to be married. The PA laws were specifically designed to accommodate Quaker marriage rules. But in some other states, you could only be recognized as married if you filled out a license and were married by a person authorized to perform marriages--a religious figure or a judge. But doing so conflicted with Quaker marriage practices. So it wasn't exactly that Quakers couldn't marry--not like the way that slaves weren't allowed to marry, or a black and white couple were not allowed to marry. It's that Quaker marriage did not fit in with the recognized ways to marry that many states had. Quaker marriage was purely a religious ceremony, and having a legal contract conflicted with that, so their marriage may not have had any legal standing. It's kind of an odd situation, since while states provided measures for both religious and non-religious marriages, I guess they did not have provisions for people whose religious laws conflicted with both secular and non-secular marriage practices. The government sees marriage as a legal contract, while it is simultaneously a religious rite. So your friend is right that it's a bit of a conflict. FYI: religious marriage and legal marriage are in many ways separate today. People can have a religious marriage ceremony but never fill out the paperwork, and therefore may not legally be married. But they may be married even so, since most states have had common law marriage statutes, according to which a man and woman who have lived together for a certain number of years are considered to be married, even though there was never a legal ceremony or contract. So Quakers, even if they didn't want to have a marriage ceremony, would still be common law married, and a common law marriage came with the same recognition as a church marriage.

FYI: relatively recently (in the last 20 years) Pennsylvania changed their marriage laws, because people were using the lax marriage laws to do things like what happened when I was in middle school: a teacher married his 13 year old student. That marriage wouldn't have been legally possible based on regular marriage laws, since a person couldn't marry until 16 or 18 or so, but as a common law marriage there was no age limit, so the marriage was recognized and the teacher wasn't prosecuted for molestation, though he did lose his job. So basically, a law designed to accommodate Quaker religious beliefs ended up being used as a loophole by the unscrupulous.