Yes butter is a thickening agent.
When a recipe calls for a thickening agent and you have butter you can use butter as the thickening agent.
Other thickening agents are things such as lard, cooking oils, vegetable oils and polysaccharides (starches, vegetable gums, and pectin), proteins (eggs, collagen, gelatin, blood albumin) and fats.
For most recipes butter makes an excellent thickening agent.
Butter is one of the healthiest ingredients that you can use for a thickening agent especially homemade butter.
Some other things that are great thickening agents in food are all purpose flour, cornstarch, arrowroot or tapioca.
I use butter in most recipes that need a fat added as a thickening agent.