I wanted to leave THIS here for reference and to let everyone reading know that NOTHING about the Revised Ranger Unearthed Arcana has been tuned or explained as it applies to Multi-Classing yet, as stated by Crawford. A single dip into this current ranger build allows considerably powerful combinations, such as permanent sneak attack.
That being said, and knowing that we have no SPECIFIC rulings yet on how to Multiclass this 'still in testing phase' Ranger, what follows are the rules for Multiclassing and how they apply to ALL Class Features. Of this there can be no argument until we hear details on how to Multiclass the Enclave Ranger.
Class Features only apply when levels in that Class are gained.
First; both of these Features are exactly that. Ranger Features; and as such the extra Hit Die and the ASI to your ranger companion do NOT work if you gain levels in other classes. Read this as 'For every Ranger level after 3rd'.
Example: Rogue Expertise
"...at 6th level you can choose two more of your proficiencies to gain this benefit..."
When you are a rogue you gain Expertise at 1st level. You don't then get to choose a second set of skills to gain expertise in at 6th level, unless you have 6 levels in Rogue. Even though it does not say as much explicitly, it is implied that this is a Rogue Feature, and applies only to Rogue levels.
The Feature you are asking about is Companion's Bond, and appears on Pages 5-6 of the Revised Ranger Document found in the Unearthed Arcana section of the D&D website. Since this is a class feature unique to the Ranger class Beastmaster Enclave, its rules only apply when you gain levels in Ranger.
Second; Ability Score Improvements (ABIs) are also class specific Features. So if you gain your ASI from the Ranger Ability Score Improvement Feature at the Ranger levels shown, your Ranger beast companion benefits from it as well. What this does not mean is that if you gain levels in, say, Fighter; which has more ASI granted to you; that it applies to your Companions Bond Ranger feature. Think of this as, if you are learning the art of the Fighter on the side, any experience you gain as a Fighter is in exchange for growing better as a Ranger and working better with your Animal Companion. Your Animal Companion is only ever linked to Ranger.
Same goes for any Features found in the Ranger list. Only Ranger levels. This can be backed up more solidly by looking at PHB pg. 163 - 164. The Multi-classing table has a definitive and exhaustive list of things that are gained when multiclassing, and points out that anything not listed there, or explicitly tied to Character Level elsewhere (such as Proficiency Bonus), is never tied to Character Level.
This makes a few solid points
- You gain Hit Dice and Hit Points only from your Class Level.
- Proficiency Bonus is based on Character Level.
- Only when you gain a level in a particular class, do you gain features from ONLY that particular class; both from any features given to you at that class level, and features that grow more powerful as you gain levels in that class.
The only exception to this rule, because 'specific beats general' is when it comes to your Animal Companion's proficiency bonus, as stated in the separate box on page 6. It says as you gain levels and your proficiency bonus goes up, so too does your animal companions proficiency bonus. It will always match your own proficiency bonus no matter what Multiclass levels you might have.