When we invest for the long term in the stock market/etfs, are the companies, who gives dividends, your asset or... are we the asset ourselves for them?

It is your asset if you own the shares in your name as a shareholder. Every shareholder owns a small part of the whole company. You even get to take part in voting during AGM.

Personal opinion we are kind of assets and owners because we give them cash in exchange for part of the company. They can then use the cash to make their business bigger and better. Closer word for it is probably Owner`s Equity