Why does it appear that judeo-christian sects avoid referring to the god they worship by his name?

I don't disagree with any of the other answers here, but let's also look at your metaphor.

When [the fish] discuss [the one Person they worship] though, for some reason [they] call him 'Person', rather than 'Yahweh'. This is strange, confusing and seemingly grammatically incorrect.

No it's not. It's perrfectly grammatical and not confusing at all, since there is only one Person.

Looking at a real world example, British people usually refer to "The Queen" rather than "Elizabeth" or even "Elizabeth II". It's generally considered more respectful. It's also less likely to be misunderstood. Americans often say "The President" in formal circumstances and when they intend to be respectful, rather than use his name. There are lots of other cases. Christians are simply following normal practice.

The metaphor fails because, for classical theists, God is not just another being, that even though infinitely perfect, powerful, wise, etc., is, in the end, in the same plane as created, limited beings. Rather, God is the absolute fullness of Being, Subsistent Being Itself. Thus, God is not an individual within a species, but is infinitely above the categories of contingent beings in a category of his own, in which He is the only one (Deut 6:4) not because it just happens to be that way, but because it is intrinsically impossible that there be another.

So, using your terms, the case is definitely not that "'God' is the kind of thing 'Yahweh' is." This was expressed in the Old Testament by the fact that YHWH was usually not referred as just "God" (El) but as

• El Shaddai, God Almighty,
• El Elyon, God Most High, and most often as:
• Elohim, literally "the gods", meaning that the only true God concentrates all the power, wisdom, etc. that polytheistic religion distributed among several gods.

It became Jewish practice to substitute Addonai (which means "lord") for the proper name of Yahweh. They generally did this because they felt at the time that the proper name of God was too sacred to actually use, the practice of which became known as the Ineffable Name Doctrine (more info here )

They also felt that since there was only one true God, the name Elohim (which means God), is sufficient. It should be noted that "Elohim" is originally a plural word, but in the context of the use of the Hebrew people, was intended to be representative of a singular God.

Addonai (Lord) and Elohim (God), translated, become our standard naming references in Christianity today, although Yahweh is also used, particularly in song.

But perhaps most importantly, Christians believe that God judges our hearts, not our grammar, and knows whether or not we seek him, specifically:

"I the LORD search the heart and test the mind, to give every man according to his ways, according to the fruit of his deeds." - Jeremiah 17:10 ESV

Some Christians regularly do use the "name" of God. Notably, the Jehovah's Witnesses call God "Jehovah", which is the name God gave himself on the Mountain when talking to Moses. I've also met plenty of non-denominational Christians that regularly call God "Yahweh", and it is in some of their music too; they certainly don't shy away from it.

Also, most Christians believe Jesus is God, and they can't get enough of that name. In this context, they would ask "Are you familiar with Jesus"? Saying "Jesus" is saying God's name. Often they feel the need to be far more poetic than a simple name, however, saying things like "Are you leaning on the everlasting arms"? In prayer, and in conversation, "My Father" is probably the most common name, likely because it is natural feeling, God is referred to as our heavenly father in many places in the Bible, and it highlights a intimate relationship. Further, though most Christians would identify God as "a person" they certainly would not equate God with personhood as you've done in your example. God is not human, nor is he even made of the same substance as people. So your contention seems one more or grammar rather than theology. On that point, the answer is simple: God, at least in monotheism, is a proper noun because there is only one God.