“the owner of matnos kehunah has a Tovas Ha’Na’ah to give it to his chaver the kohen. Meaning, it is not totally ‘hefker’. The matnos aniyim are also not totally hefker”
Oy Rebitzin, I am not sure about the Kugel in shul. but seems like you are making a kugel out of the both. (though your comments are excellent comments – your mind is certainly on track!) מתנות כהונה is considered ממון השבט and therefore has a diff status as you pointed out above, however מתנות עניים is diff – it is considered הפקר for Aniyum
see Radvaz פ”א הל’ מתנות עניים הי”א
A major difference is a קטן; a katan can be zochah Matnos Kehunah מדאורייתא whereas, מתנות עניים not. See B.M. 12a. The Ketzos explains that by Matnos anyim the pasuk states: ולעני ולגר תעזוב אותם which seems to imply that it should just be left as hefker, whereas by kehunah…(see siman 343:4) therefore even a shotah can be zocha matnos kehunah…