Hi there! Bit of a small question, I'm guessing the answer is "yes", but I don't want to assume.

So we all know they really upgraded the Ranger with Tasha's, and the Primal Companion ability in particular; allowing a bonus action to have them make an attack instead of a standard action for example. But do the other abilities that a Beast Master would normally get still apply? For example, do you still get Beastial Fury (aka 2 attacks for your beast instead of 1) at lvl 11? Do you still get to Share Spells at level 15?

And what about Exceptional Training (at lvl 7)? Does that becomes basically useless outside of the attacks now counting as magical, since you can already use a bonus action to command/Disengage/Dodge/Help per the Primal Companion ability?

Correct, the Primal Companion replaces the 3rd level feature only.  All of the other subclass features still apply.

Exceptional Training does become redundant, apart from the magical damage buff.

From the feature description:

Primal Companion

3rd-level Beast Master feature, which replaces the Ranger’s Companion feature

So I'd say it just replaces the 3rd-level feature, and you'd still get the other Beast Master features at their respective levels (even though Exceptional Training is less useful than it would be, their attacks counting as magical is pretty good.)

Quote from Aineyn >>

Hi there! Bit of a small question, I'm guessing the answer is "yes", but I don't want to assume.

So we all know they really upgraded the Ranger with Tasha's, and the Primal Companion ability in particular; allowing a bonus action to have them make an attack instead of a standard action for example. But do the other abilities that a Beast Master would normally get still apply? For example, do you still get Beastial Fury (aka 2 attacks for your beast instead of 1) at lvl 11? Do you still get to Share Spells at level 15?

And what about Exceptional Training (at lvl 7)? Does that becomes basically useless outside of the attacks now counting as magical, since you can already use a bonus action to command/Disengage/Dodge/Help per the Primal Companion ability?

This is involved.  Let me walk you through it.

1. Level 3:  This is very straightforward. 
2. Level 7:  Depends on your DM.  As a reminder, the first sentence doesn't matter - you already get this at level 3.  But what about the second sentence?  Well, 5E doesn't use tightly worded rules or grammar, so it depends.  I'll show you the two rulings.
   1. Original:  Beginning at 7th level, on any of your turns when your beast companion doesn’t attack, you can use a bonus action to command the beast to take the Dash, Disengage, or Help action on its turn. In addition, the beast’s attacks now count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.
   2. The reason for the DM to say yes, this works with the Primal Companion:  Beginning at 7th level, on any of your turns when your beast companion doesn’t attack, you can use a bonus action to command the beast to take the Dash, Disengage, or Help action on its turn. In addition, the beast’s attacks now count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.
      1. In this interpretation, Exceptional Training works on "the beast".  Your Primal Companion is a Beast, so you get this.
   3. The reason for the DM to say no, this doesn't work with the Primal Companion:  Beginning at 7th level, on any of your turns when your beast companion doesn’t attack, you can use a bonus action to command the beast to take the Dash, Disengage, or Help action on its turn. In addition, the beast’s attacks now count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.
      1. You don't have a beast companion - in fact, you explicitly don't have a beast companion, because a beast companion is what the Ranger's Companion grants, and you don't have that ability.  You have a primal companion instead which does have the type beast, but by that specious logic you could buy a mastiff in a shop - it would both be unquestionably a beast (per its type) and your companion (that's literally what mastiffs are for).  It's clear from context beast companion means the specific beast companion Ranger's Companion grants - which you haven't got.
3. Level 11:  There's no question you don't get this.  Unlike the L7 ability, where the only sentence you cared about ambiguously said "the beast", there's only one sentence here, and it refers to your beast companion, which you don't have.  But listen, Tasha's RAW is a mess - almost every class has a subclass in Tasha's where the RAW is fundamentally broken in one way or another.  In fact, Order Clerics who take Blessed Strikes - a TCOE subclass taking a class ability introduced in TCOE - experience essentially the same issue where their L17 ability does literally nothing.  Talk to your DM and plead with them to fix this.
4. Level 15:  See 11.  You don't get this and only a very mean DM will actually enforce the RAW.

List of subclasses with broken rules in TCOE - not exhaustive, I'm just listing one per class alphabetically:

1. Artificer Artillerists have a rule that requires them to cast a spell through an object, which is undefined.  This same issue later infected Spirits Bards in Van Richten's, and to this day no one knows what those words even mean.
2. An Order Cleric with Blessed Strikes can't use their L17 ability, which absolutely requires Divine Strikes.
3. Rune Knights don't gain weight when they enlarge, which is so entertaining people have worked out the necessary rules to make a halfling lighter than helium so you have your own halfling balloon to fly into battle.
4. Mercy Monks manifest a mask from nothing when they reach level 3, which is fine, but the rulebook states the monk often wears it when using the features of the subclass, which is false.  The mask does absolutely nothing other than entertain the table when you make your DM try to explain where it came from.
5. We just discussed Beast Masters.
6. Rogues are a special case - both TCOE subclasses have rules that work, they just scream that they don't work intentionally:
   1. Phantoms get 2 Wails per day at level 3, 3 at 5, and infinite at 9 because they fail the Bag of Rats test.  This is sus.
   2. Soulknives have absolutely no legal way to use their soulknives off-turn - they can neither murder someone with them as a Readied action, nor as an OA, nor anything else you can think of.  This is highly sus.
      1. BONUS ROUND:  Soulknife soul knives shimmer, which has two nearly opposite definitions and no clarity in the rulebook: both mirages and stars shimmer.  If soul knives look like stars - and there's no reason to conclude they don't - then Soulknives are forced to emit light in order to function, making them even worse assassins than Assassins are (which is a very low bar).
7. Aberrant Minds and Clockwork Souls have the same problem Sidekicks do: no-one knows if your very first level is gained or not, so rules that trigger on gaining a level (for these two sorcerers, it's their ability to swap a subclass spell out for another) may or may not trigger when you're a Sorc 1.
8. Scribes get an ability based on the "damage type of a spell", which doesn't mean anything - a spell can deal multiple damage types, like Ice Knife, and things devolve quickly when you start looking at spells that are more and more divorced from how they deal damage, like Dragon's Breath, Faithful Hound, or infamously, Animate Objects.  It's just fundamentally undefined to claim that a spell intrinsically has "its damage type" (verbatim from the book), meriting inclusion in this list.

List of classes whose subclasses stick to actual rules (balance is a separate conversation):

1. Barbarian (Beast Claws are potentially not worded as intended - we periodically get people on here upset at how many attacks one can make at level 5)
2. Bard (Creation Bards work fine, they're just clunky - when your Inspiration target uses your Inspiration for something, you have to interrupt them, even if they've split the party and you have no IC knowledge the inspiration has been used, to tell them what secondary effect happens)
3. Druid (Wildfire Spirits at L6 are potentially not worded as intended - we periodically get people on here upset that Produce Flame is so iconic but can't be channeled through the spirit)
4. Paladin
5. Rogues are being listed here as well because as I mentioned above, their rules aren't word salad, they're just really, really sus.
6. Genielocks (all 4 of them) have no apparent mechanism for not getting access to infinite Simulacra with infinite Wishes, because Simulacrum has always let the clone regenerate Mystic Arcana but now Wish is a Mystic Arcanum so it can be Simulacrum as well as itself.  That's not sus enough to make the list above - the problem is that Simulacrum is worded incompetently, not Genielocks.

So that's either 8-5 or 7-6 (depending on if you count Rogue or not) - biased in favor of word salad.  So don't let Tasha's get you down.  Explain to your DM, and ask for sanity.