If I buy and download a game through second Nintendo account do I own it on my main account too?

If the accounts are on the same switch, yes, you can play it on your main account (on that switch) but the game will be owned to the other account although I cannot see why this would be a problem unless you owned another switch. If you were to log in on your main account on another switch, you would not be able to play doom unless another account on that switch had doom (eg. Your other account).

As Steve-O's answer details, a game that you purchase on a second account will belong to that separate account - as far as Nintendo is concerned, the second account, with a separate email, is an entirely different person with their own wallet and own set of purchased games, and the two accounts can only share games using Nintendo's limited device ownership rules.

However, there does exist a compelling reason to do this despite these rules: regional accounts. Quite a few players (usually enthusiasts) in North America or Europe will create a Japanese account in addition to their home account in order to play games that only release in Japan, or that release in Japan first and in their localized region later. The different languages' copies belong to separate accounts, of course, but as long as you only use the alternate region account for games that need it, it shouldn't be much of an issue.

This can be helpful in your case because it also affects ratings and parental control limits. While Doom 3 is rated PEGI 18 in Europe, it is only rated M for Mature in North America, which places an age restriction of 17+. Therefore, you could potentially play Doom 3 by creating a North American account and purchasing it through there.