I think shooting in every direction makes more sense in a game where enemies approach from every direction. In a game where enemies are always moving coming from the right, I'm not sure it adds much to the game to be able to control the angle of your shot.

I recommend working on the enemies a bit to get a better feel for how the game will play. Shooting forward probably didn't feel very good for you because the enemies fly straight across the screen really fast. When you only shoot forward, it doesn't feel worthwhile for the player to shoot enemies down when they do that, because you're putting yourself in harms way for nothing. When enemies have more interesting movement patterns and attacks, you can get interesting combinations of threats where it's worth while for the player to put themselves in a dangerous spot to take out an even more dangerous threat. Once some of these are in place, you may have a better idea of how you want the gun to shoot.

The thing is, there are a lot of space sidescrolling spaceship shooter games like this. But honestly, I didn't see anyone with a 360 degree shooting yet.

This could be your difference, standing out and going a path none of the others didn't go yet!

Another idea would be, that the shooting is by default straight forward, so if you aim updwards, its gonna shoot updwards. The moment you stop pressin upwards, it falls back on straight forward.

There are two interacting aspects: player-controlled movement and player-controlled weapon aiming, leading to many different possibilities:

• Free movement, fixed ship facing, fixed weapon bearings (not necessarily forward). The typical scrolling or "gallery" shooter setup.
• Free movement, fixed ship facing, self-aimed weapons (on a very varied spectrum ranging from slightly tilting forward weapons according to transverse movement, to automatically acquiring targets, to agile guided missiles). Still a typical scrolling shooter setup, traditionally with reduced damage to compensate the reduction of wasted shots.
• Free movement, fixed or irrelevant ship facing, explicit weapon aiming. Rare in scrolling shooters (e.g. Tyrian 2 players mode) because it requires two directional controls rather than one (i.e. keyboard and mouse, or a gamepad, or two cooperating players, not an arcade controller's single joystick) and possibly too much attention; common in console arena shooters (e.g. Grid Wars) with unconstrained aiming (weapons can shoot in any direction regardless of movement and facing)
• Explicit facing control, movement constrained by facing (possibly forward at a fixed speed without being able to stop), fixed weapon bearings. Typical of side-view dogfight simulators (Luftrausers is a recent example, but the genre is very old) and less common top-down dogfight simulators (e.g. the recent, turn-based Steambirds Survival). Taking care of movement paths, with longer term planning, rather than position (with much less demanding dodging) is the main difference with arbitrary movement. Actually hitting enemies tends to be the reward for a good maneuver rather than the default, due to the use of realistically small weapons that need proper aiming to work. In a bad game of this kind, turning around in place distributing suppressive fire is a valid strategy (and a very boring one),
• Explicit facing control, movement constrained by facing, self-aimed weapons. A tool to make constrained movement more forgiving (if badly aligned strafing becomes somewhat effective due to mild automatic aiming corrections) or more difficult (if weapons swerve in unwanted directions or automatic target selection needs to be controlled by managing the ship's position).
• Explicit facing control, movement constrained by facing, explicit weapon aiming. "Realistic" only in unusual contexts (tanks, large ships), and likely to be too easy because hitting enemies is easy regardless of movement. Difficulty could come from a combination of serious obstacles (both movement hazards and cover) requiring clever movement anyway and demanding targets (fast-moving relative to bullet speed) requiring good aiming.

It goes without saying that a game can combine different weapon and movement control styles, including unusual ones (automatically aiming for, or strafing around, a boss, following rails, activating special-purpose weapons, etc.)