To answer all of your questions ...
I wanted to find out beforehand if I'm doing it right.
- If you are all having fun, then you are doing it right.
- If you are seeking to be more closely "RAW" aligned, not as much.
Are there any official rules for taming wild animals?
Not as such, beyond the Beast Master feature of the Ranger and a few spells (see below).
Should I continue what I'm doing already?
Are you having fun?
If yes, and all at the table enjoy this, then why stop? I can see no reason to.
Is anyone at the table not happy about this?
If yes, then maybe make an adjustment.
A few points on what the player has; a free spell effect.
What you have provided to this player is more powerful than some spell effects, though it doesn't seem to be as strong as dominate beast.
- Animal Friendship is a first level spell. It does not give this kind of control, to "tame" a wild animal. The beast is charmed, and understands that you mean it no harm. It lasts for a day. If all that your character is doing is having a companion, then you are providing a free spell each day, more or less. Since you have stated in the question that the animal is involved in combat, then that is a more powerful ability for the character.
- Animal Messenger is a second level spell. This is closer to a "tame"
beast, in that a tiny beast will deliver a message for you. The
duration is 24 hours, and longer if higher spell levels are used. As this is narrow in focus it's only partly useful as an example.
- Dominate Beast is a 4th level spell that lets you control a creature
for a minute. It can be made to fight for your, though it will make a new save for each instance of damage that it takes.
What does the Animal Handling skill do?
For domesticated animals, it helps the character keep them calm and doing what they normally do. It is not aimed at being as strong as a spell.
When there is any question whether you can calm down a domesticated
animal, keep a mount from getting spooked, or intuit an animal's
intentions, the DM might call for a Wisdom (Animal Handling) check.
For wild animals, nothing is mentioned. What the character is now doing with animal handling regarding wild animals is well beyond that. But if you all are having fun, see above. Rules as Fun1 is a way to engage with the rules to enjoy the game.
A possible change in the RAW direction ...
If you want to change anything to be closer to the rules, I suggest that you sit down with your player and discuss it with them. You can advise them that playing a Beast Master Ranger (either straight up or as a multiclass) is their best way to have an animal companion. What you have already done is along the lines of granting a player a key class feature of the Beast Master Ranger, or something that druids and rangers achieve by using a finite resource: spells/spell slots.
If the Beast Master angle doesn't sell, they can regardless of the class being played take the Magic Initiate feat when their next ASI comes along. At that point, have them either take Animal Friendship as their first level spell (less effective, but animals may stick around) or Find Familiar (more effective; an animal is with them most of the time who can do things to help them). (Add cantrips to taste).
If they are already a spell caster, and/or have Find Familiar as a spell on their list, this gets even easier.
If someone in the party has a ring of spell storing, and another party member has find familiar, then this character can summon a familiar using that spell from the ring.
What my Warlock did: an experience based point of reference
In a now dormant campaign, I took the eldritch invocation "Beast Speech" at second level so that I could talk with animals at will. It was still on me to persuade them (persuasion/deception checks) to help me, or to not eat me, but at least I could discuss it with them as an animal NPC. It was helpful, and in one case it helped us sniff out an ambush. The invocation did not grant me a tame wild animal. Your player's facility with taming wild animals is stronger than my warlock's eldritch invocation, and the same as or stronger than the Beast Master Ranger's class feature. (Aside: since I was pact of the chain, my familiar took on the more difficult tasks for scouting and keeping watch).
1 Rules as Fun {Jeremy Crawford}
RAF. Regardless of what’s on the page or what the designers intended, D&D is meant to be fun, and the DM is the ringmaster at each game table. The best DMs shape the game on the fly to bring the most delight to their players. Such DMs aim for RAF, “rules as fun.” We expect DMs to depart from the rules when running a particular campaign or when seeking the greatest happiness for a certain group of players. Sometimes my rules answers will include advice on achieving the RAF interpretation of a rule for your group. I recommend a healthy mix of RAW, RAI, and RAF!