How does your opponent verify that you're able to cast a Companion card?
Presenting a card as your companion requires you meet the companion condition. Like every keyword in magic, there is a section of the comprehensive rules that cover the ability, and those rules are what really matter. I think what you're doing is reading the reminder text as if it were the entirety of the rules for companion (reminding that you could originally cast a companion from outside the game once, now can add it to your hand for 3 mana) Here are the sections, with my emphasis:

103.1b If a player wishes to reveal a card with a companion ability that they own from outside the game, they may do so after setting aside their sideboard. A player may reveal no more than one card this way, and may do so only if their deck fulfills the condition of that card’s companion ability. (See rule 702.138, “Companion.”)
702.138a Companion is a keyword ability that functions outside the game. It’s written as “Companion—[Condition].” Before the game begins, you may reveal one card you own from outside the game with a companion ability whose condition is fulfilled by your starting deck. (See rule 103.1b.) If you do, once during that game, you may play that card from outside the game.  Once during the game, any time you have priority and the stack is empty, but only during a main phase of your turn, you may pay {3} and put that card into your hand. This is a special action that doesn’t use the stack (see rule 116.2g). This is a change from previous rules.

These two connected rules, with one being a restating of part of the other with a different focus (one from the start of game rules section, the other from the companion section) explain how companion really works - that the deck MUST meet the companion restriction for you to reveal this card. The text in brackets on the card (the reminder text) originally told you what you can do with your companion, before the rules change, but the rules themselves say how a card can become your companion. Revealing a companion that your deck does not meet the conditions for makes it an illegal deck.
As for how you verify if a deck is legal under companion rules, this works the same way as any other deck construction rule, that is you assume the deck is legally constructed until you prove otherwise. If you see a 5th copy of any of the cards (other than the 16 that can legally have more than 4 copies of) you see the deck is illegal and act on that. So if you see a 2 drop in a deck with Keruga, the Macrosage as the companion, you know the deck rules have been violated. This is actually pretty hard to hide - You can't play that card, it can't be in your hand when you get hit with something like Thoughtseize, it can't be milled from your deck, and your opponent can't play any card that lets them look at your deck like Surgical Extraction. An invalid companion would be just as hard to hide as a 5th copy of a card to an opponent paying attention.
Any potential psychological advantage fooling your opponent about what your deck is by announcing a companion that couldn't legally be your companion really doesn't exist - now you're playing with a deck that doesn't have 60 useful cards in it, it has say 50 cards that match the restriction and 10 dead ones that show you to be cheating so you could never let your opponent see them. It's also just poor sportsmanship to try and cheat.