If you are comparing dairy/eggs with seafood exclusively in terms of their total protein content, you are almost always going to do better with seafood than dairy/eggs when trying to increase your protein intake.
Unlike our WHFoods Seafoods—which account for 6 of our Top 11 WHFoods in terms of protein richness—none of our WHFoods dairy foods or eggs rise up into the Top 25 for this nutrient. Tuna is the most protein-rich of all WHFoods, providing over 30 grams of protein in one 4-ounce serving. Our least protein-rich seafood are sardines, but they still provide 22 grams of protein in every 3.2-ounces. All of our seafoods qualify as very good or excellent sources of protein!
By contrast, even though all four of our dairy/eggs qualify as good sources of protein, none of them qualify as either very good or excellent sources. Yogurt shows up as our most protein-rich dairy food, and it provides 8.5 grams of protein in one cup. Cow's milk—which provides about 4 grams in 4 ounces—is very similar to yogurt in terms of protein richness.
However, one thing to consider when deciding about dairy/eggs as a protein "booster" is the less liquid nature of cheese in comparison with milk, yogurt, and eggs. This less liquid nature means that smaller amounts of cheese are needed to provide meaningful amounts of protein. Since cheeses can provide about 7 grams of protein in one ounce (two tablespoons), a couple of tablespoons of shredded cheese on a salad can provide nearly 15% of our WHFoods daily minimum recommended protein (50 grams).
We often get asked about the issue of "protein quality" and whether a particular food can provide the type of protein that needed. For the most part, questions about protein quality refer to the amino acid content of proteins, and whether a food provides the full diversity of amino acids needed to meet our needs. You can find detailed information about protein quality in both our Amino Acids and nutrient profiles. As a summary for this Q & A, however, we would note that both seafood and dairy are fairly strong across the board in amino acid content, and you can't really go wrong with high-quality protein from either group. However, we would give the edge to seafood in this regard, due to its greater concentration of sulfur-containing amino acids like cysteine and methionine.
For more information on the subject of vegetarian diets, please see our overview article "A Practical Look at Vegetarian Diets" as well as the following Q+As.
To see the research articles we reviewed in the writing of these articles, see here.