Sweden and Finland are pretty alike, but Sweden's system has a bit less political interference compared to how it's here in Finland.

The ruling Social Democratic Party has strong ties with the unions though.

They have traditional strong ties to one subset of the unions, the LO ones. But it's complex: LO is often openly critical of the (ruling) Social Democratic Party. At the same time a large chunk of the actual union members are voting for a different party, often one in opposition to the Social Democrats.

Yeah, my experiences with Swedish unions is definitely a bit dated, but I guess SD's growth is messing things up a bit, especially in LO?

It is. The LO union members are the ones with the most to loose with a workforce surplus that comes with more immigration (really refugee asylum, traditional immigration is on about the same levels as ever). The Alliance (center-right, when it existed) recogniced this and made the traditionally left humanitarian stance their own. This backfired in a spectacular way when the alt-right wave swept the world. SD takes voters from both blocks.

>each political party has a 'favourite' union who they influence through unofficial channels.

its the other way around.

Good correction - although I would say its more or less a two-way influence system.