Thiel suing Gawker has is absolutely related to his Trump support. Trump is the presidential candidate who wants to "open up those libel laws" to make it easier to sue the press. It is certainly notable that one of Trump's prominent supports has pushed the boundaries of existing laws in the service of similar goals.
In the other direction, Thiel's Trump support suggests that his funding of Gawker lawsuits isn't just about Gawker and his history with them (which is how many people defend Thiel's actions, describing Gawker as an unusually bad actor that is reaping what they sowed). Instead, it suggests that the Gawker lawsuits are indicative of his broader attitudes about how the press should behave and what consequences there should be for press that doesn't.