With the Text/Autotext Label you can both access object properties and insert custom text. If you select this label, and in the 'Text Label' drop down menu select "Clear Text", when you click on you object it will prompt you to enter the text. Clicking on the AutoText icon will give you access to to not only Project Info, but also Object Properites, as shown in the attached image "AT02"
I drew a quick concrete beam, labeled it with information from the properties (with some custom text for ":" and "x"), and labeled the object. Then I drew a wood beam, eye-droppered the label settings and clicked the wood beam, and the label took the correct properties from the wood beam. See image "AT03".
I think this does what you are looking for, but perhaps I misunderstood. At any rate, it's really the first I have experimented with this label, and I know I am going to use it!
ArchiCAD 22
Win 10