Adapter for RecyclerView with supported onClick handling and select color text on last clicked item
I like nitwitting about kotlin, therefor only feedback about the language.
single expression functions
If your function exists out of one expression, you can write it easier.
Instead of having to write:
override fun getItemCount(): Int {
    return items.size
}

you can write:
override fun getItemCount() = items.size

code duplication
Try to remove as much duplication as possible.
if (audioRecordUI.isActive)
    itemView.name.setTextColor(itemView.context.getColor(R.color.grenadier))
else 
    itemView.name.setTextColor(itemView.context.getColor(R.color.black))

In the code above, you do three things:

Choose a colorId
Get the matching color
set the itemview to the color

Actions 2 and 3 are duplicated.
With a simple variable, this code becomes much clearer.
val textColorId = if(audioRecordUi.isActive) R.color.grenadier else R.color.black
itemView.name.setTextColor(itemView.context.getColor(textColorId))

scoped functions
When you need to reuse a variable multiple times after eachother, you can use a scoped function, eg also and apply.
instead of writing:
itemView.select.isChecked = !itemView.select.isChecked

you can write:
itemView.select.also { it.isChecked = !it.isChecked }

or even:
itemView.select.apply { this.isChecked = !this.isChecked }

And as you don't have to write this:
itemView.select.apply { isChecked = !isChecked }

Personal choice
I personally don't like the three lines for a simple if:
if (!isSelectableMode) {
    clearSelect()
}

If it's this short, I would place it on one line.
If it would be a bit longer, two lines without brackets.
if (!isSelectableMode) clearSelect()