I can certainly see the inspiration of Celeste sprinkled in this game. You got the base, but here's some points I will point out to return the favor on your constructive comment in our game! Mainly from art because I'm an artist:
-You got a decent looking title page in itch.io, so maybe put some use of that title logo and make a short title splash. Maybe not a title screen if it's too much work, but you can put it for like 5 seconds then fade out into the game.
-The game start with no tutorial, a simple tutorial in game would be nice even if it's just the button prompt and no explanation like what most Nintendo game does. Having to scroll to your description breaks immersion.
-The default audio is too loud and there's no way to tune it lower other than adjusting directly from my computer. Would be a nice little QoL to have, since I was in VC when I'm checking out games to rate one by one and the sudden blast gave me a shock.
-That being said, I don't think I find any pause button?
-The title of the game is Frostbite Mountain, but I don't even see any shape indication to that. It could be helped a little with the character wearing some parka or earmuffs, since he had to be a superhuman to survive in a harsh climate.
-Try to look for more references for sprites and animations; it's a bit weird why the character suddenly open his mouth and doing a split when he's jumping. Try to translate what the character is doing into the movements too: for example when he's holding into a wall, he should LOOK like he's holding into the wall. Try to take a look into "squash and stretch" and what it means. You can try making his face/body partially gone into the wall than having the entire body stay in shape, it won't look weird I promise! On the topic of walljump too: the character is facing the opposite direction of the wall, and it gave the wrong idea to the brain of player on what the player is about to do(it's called "anticipation"). When I saw the sprite, I expected he character to get a boosted jump into the opposite side rather than going upwards. If the player is supposed to go upwards, try making the sprite facing up instead.
-There was no clear indication on where to go first, and a good level design should allow player to tell exactly where they're supposed to go and plan ahead for their movements. Try to look at what Celeste did, they keep most of their levels to be visible in one screen and when it's not they gave you a viewer of what the section looks like so player can plan ahead before executing their jumps. I'm aware the right side is a troll when I found the ":)" but it's still not a good excuse. It might be good if there was a clear objective in first place, but the game was just booted up and there was nothing to gain and nothing to lose. Don't discourage player from playing your game, encourage them to keep going for more. This is especially important in platformers since some player who can't get into the movements and level design will drop it after a bit of frustration.
-Once again on the no tutorial, there's no way for player to find out there's a double jump unless they try. Try implementing some energy icon or color. I know some games implement double jump without notice and some people who have played many many games going to try if there's a double jump regardless, but sometimes some people won't know about it existing at all when not told. Accessibility is a strong important point and shouldn't be underestimated. See reference: In Celeste they introduce the double jump through a cutscene. It's also okay to introduce a feature slowly even though it's been accessible from the start
-No indication that the player just died. It can be an animation, a single different looking frame, or even just a straight out screen shake or transision like quick fade to black. I tried jumping down in the beginning of the game because I'm curious if it's a pit or if there's something under metroidvania style, and it was unclear if I died until I just immediately respawn. Also, I appreciate the quick respawn button even though I don't really see what's the use of it, and it's weird without any death animation.
-Might want to tweak that control a little, the character feels too fast that it's a bit hard to be precise. It's especially important in platformers that controls feels nice so player will blame themselves when they're dead instead of blaming the game. That being said, reduce spike hitbox a little. It gives player a triumphant feel even though it's fake and encourage them by making them think "I'm doing good!"
-Level design should take controls into consideration, for example there are many narrow small paths with tiny tiny jumps and it's a bit annoying to bonk my head on that section since player initially jump high.
-Tilesets should be clear on what is a background object and what is supposed to be interactable. Give more highlight to interactable stuffs and less into what's supposed to be decorative. In this case it happened to me with the light thing? I thought it was some kind of button and I tried to reach into it to find out it was nothing. Some blocks also confused me since I anticipated to pass right through it. Try to highlight the edge to indicate you can step on it.
-Give sfx on button/opening door. This one is especially important since a key item/object press need to gives the indication to player that it actually does something.
I got confused on the statue thing that makes noise, and I also can't see where the "mode" idea plays in the game. But you got the base idea of the game and I think it would be great if you want to improve more on it. Keep up the good work! :)