One's own consciousness is indeed the starting point of inquiry, but I don't think the claim that other people are conscious lacks empirical evidence (assuming a practical materialist stance here). We just don't think about the evidence explicitly, but the reasoning still happens implicitly, every time.

Suppose I look at another person and think "this other person must be conscious like me". What exactly is happening here?

First, I observe that I seem to be a human. Then, I observe that the other person is a human as well. Crucially, there are similarities between what I am and what they are. And since I am conscious, they must also be conscious, because consciousness must be caused by one of the similarities that we share, or so I assume.

This assumption can be robustly verified for humans and some animals via general anesthesia. Cause and effect (assuming a materialist stance) can be established between the drugs and human(-like) consciousness. But this is a very rough method, and it doesn't work for assessing a machine since machines can't be anesthetized.

Before anesthesia, it could still be verified in less robust ways. One way is to hit people in the head and make them pass out without killing them. This was feasible and probably commonly observed since the stone age.

But we don't know exactly how anesthetics work or exactly what in the brain is responsible for consciousness, so it leaves the door open to hypotheses that can work on machines. Here's one example: the pattern and temporal sequence in which particles such as electrons interact with eachother in a circuit dictated by an intelligent computer program may have physical similarities with how particles interact in a human brain, and certain patterns or temporal sequences of interaction may be responsible for (human-like) consciousness.

Suppose something like that were to be proven true for humans with future brain research, then it would be possible to test for human-like consciousness in any brain or computer, regardless of architecture. I suspect this is what will happen because inquiry into consciousness has always been trending downwards toward lower-level explanations, so it's probably not going to be fully explainable at the biochemical level. We'll have to go down to physics.