You can't attach a mate to one component and a mate isn't an entity that you can copy and paste., so conceptually think of a mate as a relationship-without the participating entities, it doesn't really exist.
Having said that, there are quite a few tools and techniques you could use.
If you want to copy your seat with it's mates from one assembly to another this one time, try this
Open the assembly you want to insert the seat into. We'll call this your top-level assembly.
Insert into the top-level assembly the assembly that contains the seat and place it. It's now a subassembly of your top-level assembly.
Right-click on the subassembly in the feature manager tree and select "Dissolve Subassembly." This effectively removes the subassembly replaces it with all of the parts and mates that it contained. So you'll have your seat and all of the parts it's mated to.
You can either, edit the mates on the seat and attach them to components on the top-level assembly or you can use the "Copy With Mates" tool to copy the seat from it's current location to the top-level assembly location. Then delete the lunneeded components. This technique is especially efficient if you have to copy a set of components that are mated to eachother already (like a fastener stack) because you don't have to recreate the mates with the copied parts. For example, if you copy a screw, lockwasher and flatwasher from one hole in a flange to another using Copy with Mates, then you only have to specify the hole and face of the flange not, the concentric and coincident mates amoung the washers and screw.
If you're looking for an approach to simplify mating for components that you frequently use in multiple assemblies, then you should look into creating mate references on those components. A mate reference is a framework for mating a part. You create them in the model file of the component that you'll re-use. Each mate reference consists of a selection of faces and mate types that are automatically applied when the part is inserted into an assembly and positioned over other geometry. A model can have multiple mate references each with a unique name. Multiple mate references allow you to quickly mate multiple compoents to one. For example, I have a PCB header that has two mate references: one to mate it to the PCB and another to mate the connecto to it. In an assembly, if you replace one component with another and they both have the same mate reference name, the new part will mate to the same faces as the old one. It sounds dynamite, but in truth it takes almost as much time to set up as it saves. If I were making 100K part assemblies and needed to broad part substitutions with regularity, I'd do more with it. As it is, if you're only talking about 3 mates, it probably took you longer to read this post than it would have to manually insert and mate the seat.
Coming in late on this but wanted to add what I found over at lennyworks under tips and tricks/assemblies.
In effect, this is a "copy with mates" between assemblies.
You can insert an entire assembly into a blank temp assembly model, making it a sub-assembly. Select the components you want to copy to the "other" assembly and move them out of the sub and into the top level assembly in the tree. Right click on the sub-assembly and choose replace components, and replace sub-assembly with "other assembly". Now move the components back from top level into that sub-assembly. Your "other" assembly will now have the new parts with mates intact probably mostly. DO NOT SAVE ORIGINAL or you have done a move instead!