You're actually on to something, Sven. Although a gestational carrier doesn't share a DNA relation to the baby they carry, there are some who argue that there is a "biological" connection. They don't share DNA but they shared a common environment. The medical aspects of that conversation/argument are way above my paygrade, but it's interesting to go down that rabbit hole if this is a topic that piques your interest. But in short, the argument says that there is a difference between a biological and a genetic connection. I'd love to hear a geneticist and biologist chat about this.
Actually they don't share blood the placenta prevents that. The baby has their own blood. The gestational carrier's nutrients are simply broken down into its component parts as food. Are you part meat because you eat meat?
I'm not a geneticist or doctor, but my answer is no -- the baby will not necessarily share the same blood type as the parents (sperm & egg providers), nor will it necessarily share the same blood type of the surrogate (which has no genetic connection) who carried it. Children do not share the same blood type as their parents. Blood type is determined by a combination of genetic factors inherited from both parents.