Creating a connector does not necessarily create a consumer group. COnsumer groups are not really related to the rest of your question so I'll focus answering "What is the Kafka Connect connector thread purpose?"
When Kafka Connect starts a connector, it first launches a "connector" thread. This is the entry you highlighted in the image.
The connector thread has multiple duties:
Manage task configurations. This include validation, generating configuration for each task and reconfiguration when requested.
Handle global state. The Connect runtime ensures a single instance of the Connector thread runs so it can be used to perform actions that should be done once. For example, in MirrorMaker2 the Connector thread regularly list topics in the source cluster. If it detects new topics, it reconfigures tasks to mirror it.
See the associated methods in the Connector API.
Once the Connector thread is running, the Connect runtime can use it to generate task configurations and start tasks. Task threads do the work of copying data between the external system and Kafka.