I think where the confusion is is that Photoshop calls their text and shapes "vector layers". They're not TRUE vector though. They are ALL raster-based (bitmap) tools. They just refer to them as vector because you can modify the shape or text fairly easily without the layer data getting warped, or damaged.
Illustrator tools aren't just "mainly" vector, they are ALL vector -- TRUE vector imagery -- drawn mathematically and not based on pixels.
So it's not really about the tools.. it's about the program itself. Photoshop is a raster-based software only. Illustrator is a vector-based software only. I don't think any software would have a combination of both because it would be redundant.
An easy way to tell if a tool is TRUELY a vector tool is after you use it or draw with it, zoom in .... no matter how far you zoom in to the document, the edges of the object will appear smooth -- no pixels, no stray edges, and no semi-transparent bits/pixels.
If an image is raster-based (bitmap), then you will see square pixels around it's edges and/or where two colors mix/meet.
However..... keep in mind, once you save your vector-based image in a jpg, png, gif, etc... format, the vector data gets lost because these are bitmap type of formats. That's why it's good to keep a version of the native file (.ai file), because it will retain the vector data. You can use it in the future to convert one document into ANY size and any graphic format without losing quality.