Is there literature support for the 'pop' claim that people "marry their parent"/choose a partner with traits similar to their parent(s)? I have read multiple times that we tend to marry one of our parents. Some even claim there is no way around it. I don't really see it. There are some aspects that are similar between my spouse and my parent, but since a character has very many facets, I could find some similar facets to any person.
Obviously, a person is influenced by their surrounding, especially their close relationships, like their parents. And also, obviously, we tend to stick with what we know - including people that are similar to us and what we know. But that basically just means we tend to marry someone who is similar to us and the milieu we grew up with. I don't see any need for arguing that we marry one of our parents.
What is the evidence and accompanying reasoning for the claim that we marry our parents?
Examples of such claims:

https://www.quora.com/What-do-marriage-counselors-mean-by-we-marry-our-parents-or-we-marry-at-our-level-of-dysfunction-How-can-we-make-sure-not-to-do-that
https://www.psychologytoday.com/intl/blog/tech-support/201405/why-your-partner-may-be-your-parent
https://www.tesh.com/articles/why-do-we-marry-people-who-are-similar-to-our-parents/
Yes, much literature supports this notion. Mary Ainsworth's attachment theory has a lot to say about parental attachments and how important they are for future growth; this journal article is directly related to your query. There is also related research in evolutionary psychology, see this and this.
As for why we get attracted to parental traits: the most popular explanation is sexual imprinting, which comes from evolutionary psychology: as the human species evolved, the humans who learned early on what humans looked like were more likely to mate with the same species and pass on their genes, leading to an evolutionary benefit for learning traits from your parents.