Can you have two of the same legendary, one shiny and one normal?

If you want every Pokémon both shiny and normal, no.

As other answers have stated, it's possible to get multiple copies of a legendary by trading with another player or owning multiple games yourself, or using Pokémon Bank or the upcoming Pokémon Home to store/transfer one that you caught and then restarting your game.

However, many legendary Pokémon (and some unique forms of non-legendary Pokémon if you want to include those) cannot legitimately be shiny, even if you obtain multiple copies of them. You can see a list of all of the Pokémon in question here.

Yes, this is possible.

You can trade in the different games with other (real life) people. This means that you can give them a more common Pokemon, and they give you a second (shiny) legendary. Of course, you have to find people that want to make these trades.

There is no in-game way (without trading) to get multiple legendaries.

In addition to trading, some games do allow multiple legendaries to be caught, though not the main series games.

Specifically, Let's Go does allow multiple legendaries. See this reddit thread, which mentions Serebii's Rare Pokémon Spawns page:

After beating the game and catching the Legendary Birds, the Legendary Birds will have a chance of appearing in the Sky as Rare Spawns. To improve odds, use a Lure and have a Catch Combo of 31 or more coming. They are incredibly rare so it may be a long way.

My approach in the main series games, by the way, is to buy both games - but treat one as my "primary" which holds my complete pokédex. The secondary trades up to my primary. You could certainly do this if you, for example, have X and Y, or Sun and Moon, etc. - catch the legandary on your primary first, and then go for the shiny on your secondary, trading up to your primary (or trading both to the Pokébank, if you prefer that). This won't work perfectly, as some legendaries are game exclusive, though.

If you're playing Sword and Shield, it's even easier. You can simply have two profiles in each game, one again as primary one as secondary, and then trade via a third game (either the opposite, if you're a true completionist, or someone else, or when they do finally add PokéHome via that) to have one complete Pokédex. I used this to get all 3 starters (I have 2 profiles in my Sword game, and 1 in my Shield game, each with a different starter, then using breeding to swap them about). I don't believe this is possible in previous (DS) games, without either having multiple actual games or erasing your save file and restarting, but am not an expert there. (Restarting is always an option, as mentioned in the comments to another answer, but it does require using the Pokébank.)