[RPG] Are there warm clothes in 5e

Can the GM ban references?

Yes. Specifically, "the D&D rules help you and the other players have a good time, but the rules aren't in charge. You're the DM, and you are in charge of the game." (DMG p.4, "The Dungeon Master," emphasis in original.)

That being said...

This is strange, as presented. In nigh-thirty years playing I've not run across a GM who disallowed looking up references. Not during your turn? Sure. Not at all? Weird.

(When running a table that operates under time constraints I will often skip a player who's not "ready" for their turn in combat. This tends to fall harder on spellcasters than on martial types, and harder on new players than on experienced hands. Of course, this is articulated at the beginning as a standard of play.)

But according to your edit, that's not what's going on here. Your GM seems to be basing this restriction on two ideas: the character needs some time to be able to "fully" use their class features, and the player's skill should impact the character's abilities.

1. The character can't use their class features--new spells, in this case--until they're "broken in."

   This has no basis in the rules. In short, "Beyond 1st level" (PHB p.15) tells us that when your character earns certain XP they gain a level. The class descriptions tell us what new things the character can do when they gain a level. @Kryan's answer has the right of this: the two new spells your wizard now knows by dint of increasing a level represent the work that character's already done to learn new spells, not some new task the character needs to take on.

   (As an aside, do fighters only get part of their Ability Score Improvement or feat until they've sufficiently proven themselves at their new level? Do druids pop out of Wild Shape suddenly because they're not yet well-trained in a new form?)

   Where there is some support for something like this is in the question of when a character earns XP or can gain a level. Adventurer's League rules, for instance, only allow a character to gain a level when they've completed a long rest or at the end of a module. That is, if killing goblin three of seven in an encounter would put you at a new level, we usually don't stop combat to do it then. This is also subtly achieved by GMs who award XP at the end of sessions, or by "milestoning."

   Again, though, once you've got the level, you get all the class features that come with it, full stop.

2. Player skill = character ability. Charging your character an action for you to look at your character's spells is... insane. That's a huge hit in the action economy. This seems to stem from an idea that the player's ability to memorize everything redounds to the character's ability to perform in the fiction.

   The idea that player skill should be important is an old one, and has plenty of merit to it, I think. But this is an incredibly ham-handed way to bring player skill into the game.

   (Again, I've got to wonder if your GM tries to stab the fighter-player just to see how well the player reflects DEX 18?)

"Shouldn't I be allowed to reference my spells...?"

This is hard to answer: there may be things going on that we're not aware of. What you should be able to do, without question, is talk to your GM about what's going on. Ask them what purpose they see their rule serving. Ask what they're hoping to achieve. Describe the difficulty it's causing you. Ask them for help playing your character well at their table.

(This all assumes the very best of your GM: that there's a good reason, poorly articulated, for this and that they're interested in helping you play your character. I hope that's your situation, rather than the other one: you're sitting at a table with a petty tyrant who hates spellcasters and is actively trying to make play difficult for their players. In that case I suggest you find a different table.)

1. Characters are free to pick any eye-color they want. The character can have gray eyes just because the player says he or she does.

2. Likewise, having seen an angel in the past is, in most campaigns, a reasonable thing for a player to include in a character’s backstory. In some campaigns, for example in a campaign where angels haven’t been seen for eons, the DM might want to nix that backstory concept, but in most D&D campaigns angels aren’t that rare.

3. The best source of information, therefore, on why this character has gray eyes due to an angelic encounter, is the player him-or-herself.

Basically, this sounds much, much more likely to me to be the player inventing a concept for their backstory, than it seems like a reference to any specific feat, condition, or effect in D&D. Actually, if it is from some source beyond the player’s imagination, I’d suspect a TV show, anime, comic, or similar, before I would suspect something out of D&D itself. I cannot find any mention of such a thing (though admittedly, attempting to search for information like this is rather difficult, since the results have lots of things about gray angels, seeing angels in real life, photos of pretty gray eyes, etc. etc.).

Therefore, just talk to the player about it. Ask them where, if anywhere, they got this from.

If the player is just making this up, then without a particular reason to not do so, I suggest just letting him or her run with it. The player has made up a detail about the world: that seeing an angel, at least in some circumstances, can cause someone to have gray eyes, and this is a thing that happened to the player. Players adding to the world’s detail is a good thing: now your world has a little more going on, and you didn’t have to do the work.

If the player is referencing some non-D&D material, I would be somewhat more leery—D&D tends to model other narratives poorly, and trying too hard to bring a non-D&D character into a D&D campaign is, in my experience, rather problematic. Characters inspired by characters in other media is fine, generally, but gray eyes due to an angelic encounter is really quite specific. I would be somewhat worried about the player trying to warp the game more and more to match whatever media he or she is referencing. I dislike it when players try to do that in games, whether I’m DMing or one of the other players.

If the player is referencing some D&D material, then you have less concern, and more opportunity. They can point you in the right direction, to where you can read up more about it. You can more easily judge for yourself if this is appropriate, if there is some feat or whatever that’s expected here.