This is a little too powerful

When it comes to 5th edition tactics, spreading the role of Concentration around is very effective as it allows the high Constitution characters to concentrate on their own enhancement spells like enhance ability, haste, or polymorph. This also opens up the spellcaster's Concentration for other spells (usually less tactically important ones such as entangle or cloud of daggers).

Your feat makes accomplishing this virtually trivial. You simply pack a rune with the enhancement spell for 20 gp (a relatively small amount) and the fighter/barbarian/paladin has its own Concentration spell. This is even more powerful than spell scrolls (which linkassin's answer tackles the relative monetary and time costs of) since the character need not even possess the correct spell list to gain access to the spell.

Stockpiling

This also allows the stockpiling of runes for a dangerous dungeon throwing out encounter balance. Scribing spell scrolls takes significant downtime, but a spellcaster could convert their entire set of spell slots into runes in a day at lower levels. This means that they have a full set of additional spells to use for whatever difficult challenge presented itself.

Wording

There are a few instances in your homebrew where the wording doesn't quite align with the styles that Wizards of the Coast seems to adhere to. Here is a revision I put together in an attempt to remedy this. It is by no means 100% supported, but based on observations I've made in reading through the books (probably too many times):

Rune Caster

Prerequisite: the ability to cast at least one spell of 1st level or higher

You've developed the skills necessary to set your spells in runes to hold their magic for a later time. To do this you cast a spell as normal but the casting time is increased by 1 hour, you must expend additional material components related to the type of scribing costing 20 gp per spell level, and the spell has no effect other than creating the rune. The object you are scribing the rune onto must remain within reach for the entire casting time.

While scribing the rune, you choose a trigger condition that activates the rune. This trigger condition must involve either a command word or a touch, and can include both. You can further refine the trigger to require certain characteristics such as by requiring a specific creature or creatures, or the use of certain body parts. The set of creatures or body parts that are required can be as broad or specific as you like.

When a creature activates the rune, the creature casts the spell as normal except that a spell with a range of Touch becomes a spell with a range of Self. If the spell has an area of effect, the origin of the area of effect is the rune. Once the rune is activated, it disappears, leaving no trace of its presence.

Note that I removed the wording about choosing targets since it is redundant as the creature casting the spell always chooses the targets. Additionally, the section on having the GM interpret the trigger "as the character would" is largely confusing and if the GM is actively trying to use wording to adjust the usefulness of the feat, there are probably other problems at the table. The rest of the changes are mostly to allow it to read in plain English with some small word-choice decisions based on other similar examples in the official content.

Spell Mimic is too flexible

I would not say that:

This feat is basically a watered down version of the arcane tricksters 17th level Spell Thief ability.

Because in many ways the Spell Mimic feat is better than the Spell Thief feature:

• The Spell Mimic's trigger is less specific than the Spell Thief's.
• The Spell Mimic cannot fail, whereas the Spell Thief can (and does more often than not).
• The Spell Thief's ability to reproduce a spell is implicitly limited to 4th level spells slots, whereas the Spell Mimic can even work with 9th level spells.
• The Spell Mimic recovers on short rests while the Spell Thief does not.

I'll grant you that in some respects Spell Mimic in inferior to Spell Thief:

• Spell Thief preserves the memory of the new spell for much much longer than Spell Mimic.
• Spell Mimic is implicitly useless with spells that have a casting time greater than 1 minute, whereas Spell Thief is not.
• the Spell Mimic may not use their prefered spellcasting ability, whereas the Spell Thief always uses intelligence.

But Spell Mimic also lacks effects that make Spell Thief better against foes and inefficient with allies:

• Spell Thief negates the triggering spell, whereas Spell Mimic does not.
• The target of Spell Thief forgets the triggering spell whereas the target of Spell Mimic does not.

This means Spell Mimic is best suited to reliably double the efforts of your allies instead of "using an opponent's strength against them".

In practice, Spell Mimic gives a caster temporary access to the spells prepared by the rest of the party when they are most needed and effective. This flexibility combined with some teamwork makes Spell Mimic unbalanced.

Whenever the party really needs something (whether it be heals, buffs, control, or raw damage) as long as at least one other caster can provide it, the character with Spell Mimic can double their efforts by mimicking their most suitable spell. Whereas it is much rarer for an opponent to use a spell worth mimicking in any given circumstance.

Then there are also class-specific shenanigans, like:

• A Warlock can use buffs (especially ones with long duration) and heals they usually don't have access to before a short rest.
• A Sorcerer can make up for their notoriously small amount of spells known.
• A Wizard can create contingencies of mimicked spells.
• A multiclass of two casters can mimick spells of a higher level than they can learn.
• A Tempest Cleric can spam a couple max-damage chain lightning.
• Etc.

My bare minimum recommendation to balance Spell Mimic is to somehow discourage targeting allies and limit it to spells of 5th level or lower.

However, it may be easier to achieve a feat that uses an opponent's strength against them via modifying something less convoluted than Spell Thief. I would explore an effect similar to the Ring of Spell Turning with adjustments like removing the advantage on saves and limiting which spell levels can be turned to match the growth of a half-caster.