Is a license to use of a particular asset could be considered part of its cost?

Might depend on the situation and what type of license. A one-time license could possibly be counted as part of the cost of the item, but off the top of my head I can't think of anything like that. (Like a license to do construction would be part of the cost, but that's construction and not an outright purchase. But one-time costs like closing costs on purchasing a building would count. I just can't think of a one-time license to use an asset off the top of my head.)

A license fee paid on a regular basis to use the asset would likely not be part of the cost, simply because it's an ongoing expense. That's kind of like a royalty, which is an expense. I don't see why that should be different than ongoing maintenance or insurance or anything like that, and those are not part of the costs. For instance, the license plate on your car (ongoing) is an expense, not part of the cost of the car.

I'm not totally comfortable saying that since I don't know what this license and asset actually are.

But as a general rule, all costs associated with the initial cost, getting it, installing it, even repairing it if it's used, are counted as the asset. But ongoing costs associated with using it is not the same thing.