"Put up" - a phrasal verb of 'put.' In your example, it means that Moomintrol was assembling and constructing the swing set when Sniff got home. Putting up the swing set could include attaching the support bars, chains, swing seats, and so forth. I wonder if other native speakers of English are going to jump on this well-used phrasal verb. Put up the shelter. Put up the posters. Put up with your boss. Put up your hair. Put up the cash.

in this very sentence it's not a phrasal verb it's just "put" and "up" meaning "erecting"

Oh, I love Tove Jansson's Moomin books! One thing to keep in mind is that you are reading English translations, and the translations were by native Finns. They are in good, correct English, but just a bit formal... and, as you'd expect, tending toward British English rather than US English. When I was thinking about how to define "to put up," the first thing that popped into my head was "to build." It is the part of a building process where things that are lying on the ground go up into the air. When we go camping, when we get to our campsite we put up our tent. When we live, we take down our tent. I wasn't sure if "put up" was "officially" a phrasal verb, so I checked a dictionary, and the American Heritage dictionary says it is.

"to put up" something -- to build, to hang, to construct. Moomintroll was just hanging a swing when Sniff got home. Повесить чего то. Your understanding is very correct. In this sort of context the verb "to put up" means to hang or construct a thing vertically, "to put up" with something -- to suffer through a challenge, to patiently react to your little brother, to deal with a minor series of problems. Терпеть / Потерпеть чего то

= to install / to place / to hang / to build or construct.