Is it correct to think a stock has risen 300% and it is as if it is giving out 20% dividend and never sell it?

The stock price appreciation is a "sunk gain" (analogous to a sunk cost). The relevant yield is 5% because the choice you have now is between the $2/year income stream and the $40 you'd get by selling it. Or to put it another way, if you reinvest the dividend, you have to do so at a price of $40, not $10. The original purchase price should play no role in decisions about an investment (with the possible exception of capital gains taxes).

It's a nice hypothetical position to be in but companies don't double their dividend every few years.

If non sheltered, it would make no sense to sell the stock for $40 in order to buy another stock with the same yield because there would be a large capital gains tax, reducing your compounding. It's better to defer that.

Company don't usually follow dividend yield, but payout ratio. It doesn't matter if the price is going up and down, if the net income is stagnant, the dividend will also be stagnant.

But, to address your hypothetical question, if the current price is $40, and the dividend is $2, then the relevant yield is 5%, because, if you sell your stock for $40, then buy another stock with same yield, at the same price, you'll get $2 for $40, and it's 5%.

To put it simply, your stock is now $40 with $2 dividend. There is another $40 stock with $4 dividend.

You'll gain more money if you sell your current stock to buy the 10% yield stock. So your current stock's yield is not 20%.

At this point, if 20% is something that you'd be satisfied, should you consider it a 20% dividend stock, and never sell it for the rest of your life as long as it is $40 or higher and giving out 5% dividend?

This is the issue. You have $40 stock with a $2 dividend. Your actual cost is meaningless when looking at these 2 numbers. Say, I found you a good $40 stock with a $2.50 dividend. Does the 20% figure help or hurt your calculations?

As Chris calls it "yield on cost", it's interesting to think of it this way, but not for sake of any comparison. As others will likely have a different cost.

Imagine your shares went from $10 to $40. If you sell all your shares for $40 and then buy them back immediately for $40 then nothing has changed. But now, you would be calculating the $2 dividend as 5% of $40, and not as 20% of $10.

Or let's say in the original situation, you find another share that costs $40 but pays $4 dividend (10%). The way you calculate the dividend as 20%, your $2 dividend (20%) somehow manages to be better than the $4 dividend (10%). Clearly that can't be right.

Summary: You always calculate the dividend as a percentage of the current share price, not the share price when you bought the stock.