"Appropriate level" means one with which the spell could normally be cast
The relevant quote from the new Fey Touched and Shadow Touched feats is (TCoE, p. 79-80):
You can also cast these spells using spell slots you have of the appropriate level.
In this case, the phrase "spell slots [...] of the appropriate level" means you can cast the spells using any slots that would normally be appropriate for you to cast the spells with. There is no ambiguity here, as "spell slots" is plural modifying the subject to which the "of the appropriate level" applies to to be the totality of the spell slots you have.
So you can use the base spell slot level of the spell, or you can opt to use a higher-level spell slot as outlined in the PHB rules:
When a character casts a spell, he or she expends a slot of that spell's level or higher, effectively "filling" a slot with the spell.
So, at a minimum you have to use a slot of the spell's level, but are free to use a higher one.
Is the word "the" not intended to restrict it to only using a single slot level (the base level)?
If the feature was intended to only allow the spells to be cast at the base level, the wording would need to be "You can also cast these spells using a spell slot you have at the base level of the spell." or "You can cast these spells using appropriate spell slots, but only at their base level."
What about the Artificer Initiate feat?
One of the other points raised is that in another feat in the same book, Artificer Initiate (p. 79), the corresponding part of the description says:
You can also cast the spell using any spell slots you have.
Here, it is unambiguous that you can cast using higher-level spell slots. So does this mean the text for the Fey Touched and Shadow Touched feats was intentional to prevent being able to use "any slots"?
There is a significant difference between the Artificer Initiate feat and the Fey/Shadow Touched feats. The Artificer Initiate feat only allows you to learn 1st level spells, whereas the Fey/Shadow Touched feats allow for 2nd level spells as well (e.g. misty step). Given that 5e is explicitly an exceptions-based game (via the "specific beats general" principle), if the same wording were used on the Fey Touched and Shadow Touched feats, it would create an exception allowing the 2nd-level misty step to be cast using a 1st-level slot (given that 1st-level is a valid choice from "any" spell slots, but not from the "appropriate" spell slots).
It's also worth considering, that not all feats (or indeed features) are written by the same writer, and that 5e is written using natural language, as opposed to strictly defined and interpreted language. There are multiple ways to convey the same meaning using natural language. As a result, trying to put two features side-by-side and using one feature to try disprove a feature application in the other due to wording differences won't be appropriate in most cases.