How important is specialization to branding? How would you feel if a strong dairy brand enters into the beverage sector using the same brand?

Specialization is a key part of the Branding paradigm.
However, this is an industry-specific criteria.
For example, a strong dairy brand has connotations relative to dairy products, i.e., soft, nutritious, essential, natural, beneficial, bovine.
It has a connection to living a healthy life and applies across all age groups.
However, the same emotions would not apply to an aerated beverage as the brand would cater to a different target audience for an aerated beverage.
On the other end of the spectrum, specialization would not appeal to an FMCG hygiene brand as strongly.
Mr.
Muscle offers a multitude of products across the hygiene range and the brand integrity is constant because the core proposition is the same for all products across the range, i.e., cleanliness and hygiene.

If he is using the same dairy product brand as the beverage product brand, then it will be like your home electricity technician is trying to convince you he is good as well in computer maintenance! Plus it will have the risk of failure for both products if one failed.
Some strong and strict operated companies do this "Like Nestle", but I highly doubt that is easy to keep the brand promise, that's why P&G hides the corporate brand, and uses a house of brands architecture model.

Hossam, it's interesting you mentioned Nestle because puting aside all local Iranian brands Nestle is actually making the same mistake in Iran (and the middle-east) by launching drinking water with the Nestle brand!

To answer this question you have to build a strong analysis to to find how much it's accepted from your target market in beverage For me as point of view, i prefer to build brands portfolios in FMCG industry under family brand scenario for each of your product lines, that because you can't loud your brand value proposal for booth kind of brand dairy and beverage and specially if your segment portfolio is different for example in age , except you have the same target market with the same values and this is hard and not clear, however also depend on the association between the brand you use and the product you introduce to your public target market, for example if you use brand name associate strongly with dairy and use it for beverage you will kill booth products and make message confuse