Baal has explained a lot about antibodies, I want to add some points.
1. Can a blood test tell people if they have protective antibodies (Abs) against COVID-19?
When people are infected by SARS-CoV-2, many types and kinds of Abs will be produced. Most (all?) of them are binding Abs, which bind to certain regions of SARS-CoV-2 (i.e., S protein on the outer part of the virus, N protein on the inner part of the virus, etc). However, not all of them are neutralizing Abs (protective), which have the ability to inhibit viral infection; Abs against N protein probably cannot inhibit infection because N protein on a "live" virus was not exposed and accessible. The commercial IgM/IgG Ab blood tests (ELISA run in a lab or test paper run on site/at home) can only tell whether there are binding Abs in one's blood - a positive result suggesting a current/past infection. The only way to tell whether there are neutralizing Abs in one's blood is run a several-day long experiment, to grow permissive model cells, challenge the cells with real virus (high level of safety regulations) or with pseudo virus (very low safety regulations), -/+ the plasma/serum sample to be tested, then after 1-2 days of infection, collect the culture medium and compare how much virus is released from the cells. Neutralizing Abs can inhibit viral infections and result in a reduction of viral release. When officials and "experts" say the commercial IgM/IgG Ab tests can tell who has protective Abs and are ready to go back to work, I would say that's a white lie. The nucleic acid test is required to see if people with Abs still shed virus or not.
2. Can anyone infected with COVID-19 or receive vaccine develop neutralizing Abs?
We don't know yet and researchers are testing. I believe most people do but there are always outliers. We should keep in mind that there are individual variations among us.
https://www.medrxiv.org/content/10.1101/2020.03.30.20047365v2 This preprint found that neutralizing Abs were not detected in a few mild symptom COVID patients. One can argue that this was due to the detection sensitivity of the experiment. We have to wait for more reports, especially from vaccine trials.Some people may get infected more than once if they indeed cannot develop neutralizing Abs after infection or vaccination. But that won't affect the population much as we only need ~70% (1-1/R0) not 100% of population immuned to achieve herd immunity (by infection or vaccination).
3. Monoclonal Ab (mAb) drugs vs convalescent plasma
Both mAb and convalescent plasma have to be neutralizing Abs in order to work. Many mAbs have been identified by research institutes and pharmaceutical companies. However, I don't think any one is on clinical trial yet. There are studies and news reports that convalescent plasma therapy worked for severe COVID patients and some randomized trials are in progress.
https://jamanetwork.com/journals/jama/fullarticle/2763983
https://www.pnas.org/content/117/17/9490
Is it necessary to develop mAb drugs if convalescent plasma can be collected from millions of the recovered? The effectiveness of neutralizing Abs from different donors of convalescent plasma may vary a lot (see the preprint in 2); the blood also contained non-neutralizing Abs. In contrast, mAbs are easier to do quality control and standardized during manufacture. Maybe not a good metaphor: You want to obtain Vitamin C from fruits and you can access a few kiwi and many apples, and you worry about too much sugar by eating fruits to get enough Vitamin C; then some companies come in saying, "Don't worry, we are making Vitamin C tablets".
Many US companies launched COVID-19 mAbs projects early on: on 2.12 Vir announced hits from SARS survivors; on 3.12, Eli Lilly announced hits from COVID survivors; on 3.17, Regeneron announced hits from humanized mice. There are discussions that multiple mAbs should be considered to use as a cocktail; that will lead to more complicated and expensive clinical trials. Manufacture cost of mAb is high, ~$260 per gram, but retail price can be much higher, for example an mAb anticancer drug costs $9500 per gram. It is relatively easy and cheap to identify COVID mAbs hits (unlike anticancer drugs, time and money were spent to search for novel targets), I don't see why companies won't pursue the profit.
4. Why antibodies from SARS survivors worked for COVID?
It has been reported by different research groups since Feb that antibodies against SARS (either mAbs from SARS survivors, or polyclonal horse serum which were used to treat SARS patients) had neutralization effect on SARS-CoV-2. Although SARS and SARS-CoV-2 have many sequence differences, they share 3D structural similarities. Our Abs are not that "sensitive" to identify such differences, which may be good news as the differences among SARS-CoV-2 strains must be smaller than SARS vs SARS-CoV-2. It is less concerning that the vaccine will fail soon as flu shot.
I heard that antibodies against alpha coronavirus (common cold) may also cross-react with SARS-CoV-2 (no published data though). This may explain why some people were less susceptible when they contacted the infected. But it may also cause false positive issue for Abs blood test. There are recent news reports that people tested positive for COVID Abs and thought they caught the virus as early as Sep 2019. It is hard to interpret the Ab test results without having swab nucleic acid tests, X-ray/CT scans, and other clinical investigations.