Does the Rune Knight fighter’s Giant’s Might feature affect the character’s height

The class feature description has everything you need to know.

The feature description for Giant’s Might is complete - you don’t have to look elsewhere to learn its effects:

As a bonus action, you magically gain the following benefits, which last for 1 minute:

• If you are smaller than Large, you become Large, along with anything you are wearing. If you lack the room to become Large, your size doesn’t change.
• You have advantage on Strength checks and Strength saving throws.
• Once on each of your turns, one of your attacks with a weapon or an unarmed strike can deal an extra 1d6 damage to a target on a hit.

A little bit of extra damage is baked into the feature, and these are all the benefits of the feature. You don’t have to look elsewhere to learn how this feature works.

You don't need the Monster Manual or Dungeon Master's Guide to understand your class features.

The Dungeon Master's Guide contains guidance for the DM can use to create new monster stat blocks:

If you want a full monster stat block, use the following method to create your new monster.

The introduction to the Monster Manual explains all the components of a monster’s stat block. Familiarize yourself with that material before you begin. In the course of creating your monster, if you find yourself unable to make a decision, let the examples in the Monster Manual guide you.

Once you have a monster concept in mind, follow the steps below.

The instructions for creating a monster include the following guidance for bigger monsters, which is where you got the idea:

Big monsters typically wield oversized weapons that deal extra dice of damage on a hit. Double the weapon dice if the creature is Large, triple the weapon dice if it’s Huge, and quadruple the weapon dice if it’s Gargantuan. For example, a Huge giant wielding an appropriately sized greataxe deals 3d12 slashing damage (plus its Strength bonus), instead of the normal 1d12.

This is guidance for the DM to use to homebrew a monster. This has nothing to do with player character class features. This section isn't even giving "rules" about anything. This is telling the DM, "If you want to create a monster, here's some help".

This is largely up to the DM to determine, but it's probably worth talking about the reason for that clause.

In general, sections like these are just meant to tell everyone involved what the options are: "you become large" or "you don't grow bigger". It implicitly tells the DM and player that you can't use this power to grow large and break open a coffin you're trapped in, or burst your chains, or explode a purple worm from inside, or whatever.

Both players and DMs have a long history of attempting to treat any size-changing effect as if size change were an unstoppable force, whether it's a creature doing it or an item ("I put one end of the Compliant Staff against the wall and the other end against the stone door, and command it to grow!") Worse, how to rule the power can easily become a point of contention: "I use my Giant's Might to become big and explode out of the coffin!" "Okay you become large. You are crushed to jelly against the inside of the coffin, roll up a new character." "WHAT?! THAT IS NOT HOW IT WORKS!"

As a result, size change effects generally have some kind of explanation of what happens when there's not enough space, which is usually "it fails" or "you get as big as you can given the space".

In other words, there isn't really a balance issue whether the DM decides you become large and have to squeeze or the ability just doesn't work (though I'd probably have you squeeze because we all saw Alice in Wonderland get stuck inside the White Rabbit's house and it's hilarious). Rather, the benefit is in giving a clear and concise answer about what happens when squeezing isn't a possibility.