Well, that is actually because «je le tiens de X» means literally that it is X you taught you that, and that this fact is still true and useful today. Thus the use of the present.
In fact, in your first sentence, you as well implies that your mother is still alive. If she was dead, then you should use the past tense for the second expression.
Actually, you have to say something like :

Napoléon tenait de sa mère l’art de la cuisine japonaise.

because he is dead.

The verb «tenir» is normally at the past tense outside of this expression, like :

Ce jour là, je tenais fermement sa main.

You could not say «Ce jour là, je tiens fermement sa main», as it is not true anymore.

And finally, it is perfectly possible to encounter this expression at a past tense if you refer to a past fact, which has not much to do with today, even if the protagonist is still alive. For example:

En 1970, j’eus la chance de ma vie. Je tenais d’un ami de mon frère qu’un emploi de comptable venait de se libérer dans une grande banque.

Here, I am alive, but the information is quite useless today. And here, the use of the present would be incorrect.

So in a nutshell, the present here is due to the fact you are speaking of now, and that the information taught to you is still used today. And even if it is true this expression is often encounter in present, it is not a necessity.

Answering to the last question as Vincent already answered the main one.

Is this an idiosyncrasy of the verb « tenir (qch de qn) » when it is used with this specific meaning? Or are there some other verbs that share this characteristic?

Not only tenir, this happens with all verbs describing a temporary action vs verbs describing the permanent state resulting from this action.

Je comprends l’anglais / j’ai appris l’anglais.
Je connais le maire / j’ai rencontré le maire.
Il est mort dans un accident / il s’est tué dans un accident.

Sometimes, it can even be the same verb:

Le château a été contruit au XVe siècle.
Le château est construit en haut d’une colline.

In french, we use the present to relate actions that still occurs now, event
if they started in the past:

Present perfect in english:

I have been here for two hours

But present in french

Je suis ici depuis deux heures

The first sentence uses present simply for that rule, there is nothing special in the tenir verb.

If mother was already dead, it would be:

Tout ce que je sais sur l’art de la cuisine japonaise, je le tiens de ma mère qui le tenait d’ailleurs de ma grand-mère.

But still with a simple imparfait and not a plus que parfait nor passé antérieur