Hyperspectral target detection (HTD) aims to identify specific materials based on spectral information in hyperspectral imagery and can detect point targets, some of which occupy a smaller than one-pixel area. However, existing HTD methods are developed based on per-pixel binary classification, which limits the feature representation capability for point targets. In this paper, we rethink the hyperspectral point target detection from the object detection perspective, and focus more on the object-level prediction capability rather than the pixel classification capability. Inspired by the token-based processing flow of Detection Transformer (DETR), we propose the first specialized network for hyperspectral multi-class point object detection, SpecDETR. Without the backbone part of the current object detection framework, SpecDETR treats the spectral features of each pixel in hyperspectral images as a token and utilizes a multi-layer Transformer encoder with local and global coordination attention modules to extract deep spatial-spectral joint features. SpecDETR regards point object detection as a one-to-many set prediction problem, thereby achieving a concise and efficient DETR decoder that surpasses the current state-of-the-art DETR decoder in terms of parameters and accuracy in point object detection. We develop a simulated hyperSpectral Point Object Detection benchmark termed SPOD, and for the first time, evaluate and compare the performance of current object detection networks and HTD methods on hyperspectral multi-class point object detection. SpecDETR demonstrates superior performance as compared to current object detection networks and HTD methods on the SPOD dataset. Additionally, we validate on a public HTD dataset that by using data simulation instead of manual annotation, SpecDETR can detect real-world single-spectral point objects directly.