Yes, that's how FB poses it, as part of your profile. For instance, "What music do you like?" You type in some groups, and that counts as liking them. Suddenly, you get to see news about each concert, blu-ray, etc. they're doing, right in your feed. Not a great experience.

I'm not sure why anyone would assume a very casual social "hey I like X" is equivalent to "subscribe".

People make that assumption because facebook is filled with social people rather than anti-social people who worry that someone they 'like' might actually talk to them.

In normal social circles there an expectation that the you want to hear about things you 'like'. For instance if in real life I have a friend who 'likes' classical music then when I hear something about classical music I will talk to them about it.

Also like is not like subscribe because the idea behind subscribing is that you receive everything. For instance if I 'subscribed' to a magazine I would be rather upset if they only sent me every 7th issue, or only sent me issues when the newspaper had not arrived.

"Do people associate liking things with "completing their profiles"?"

People commonly associate/link their personal identity with media and products, certainly.