The statement in another answer that common allergens (like ovalbumin) are not labelled or measured in vaccines is specific to the US. Since the Q is about the EU as well, I'm fairly certain that some testing and labelling is required or at least strongly recommended for such trace products. [For example](https://www.ema.europa.eu/en/documents/product-information/fluenz-tetra-epar-product-information_en.pdf), the EMA info for a flu vaccine says:

> The vaccine may contain residues of the following substances: egg proteins (e.g. ovalbumin) and gentamicin. The maximum amount of ovalbumin is less than 0.024 micrograms per 0.2 ml dose (0.12 micrograms per ml).

There is certainly an EMA [document](https://www.ema.europa.eu/en/documents/scientific-guideline/guideline-influenza-vaccines-quality-module-revision-1_en.pdf) that suggest this should be done:

> Process related impurities (e.g. ovalbumin for egg-derived influenza vaccine / host cell protein, residual host cell DNA for cell culture-derived vaccine, downstream-derived impurities such as reagents used for inactivation/splitting) should be identified, quantified and data used to set release specifications.

There are also country-specific guidelines like [these](https://www.bmj.com/content/358/bmj.j3468) from the UK on how to use such vaccines:

> The Joint Committee on Vaccination and Immunisation has advised that, except for those with severe anaphylaxis to egg, which has previously required intensive care, children with an egg allergy can be safely vaccinated with the live attenuated influenza vaccine (Fluenz Tetra); those with clinical risk factors that contraindicate Fluenz Tetra should be offered an inactivated influenza vaccine with a very low ovalbumin content (<0.12 μg/mL). 
>
> In adults, the ovalbumin-free influenza vaccine (Optaflu), if available, can be used in any setting, regardless of the severity of the egg allergy. Adult patients can also be immunised in any setting using an inactivated influenza vaccine with an ovalbumin content <0.12 μg/mL, except those with severe anaphylaxis to egg that has previously required intensive care.

There's also [a 2014 book](https://books.google.com/books?id=5XLCBwAAQBAJ&pg=PA105) with this summary table (which includes ovalbumin), which suggests that the WHO drew up some recommendations in 2005 and the EU adopted some (stricter ones) in 2012. No mention of US though:

[![enter image description here][1]][1]

Corroborating with another [source](https://cdn.cytivalifesciences.com/dmm3bwsv3/AssetStream.aspx?mediaformatid=10061&destinationid=10016&assetid=17500) that cites the 2005 WHO Technical Report Series No. 927, the ovalbumin limit in that table is per dose (rather than per ml as in earlier quotes).

---

On another angle, the red-underlined quote is pretty misleadingly interpreted. "Some information regarding additive and residuals is considered to be a trade secret." Sure, "some information". But what that means in practice [is more like](https://www.hhrjournal.org/2021/06/addressing-the-risks-that-trade-secret-protections-pose-for-health-and-rights/) some process details, and sometimes some clinical data (the last point is the most contentious), e.g.

>  For example, when immense public pressure led to the release of US vaccine contracts in November 2020, the public learned that the Johnson & Johnson contract explicitly allowed the company to keep secret “production/manufacturing know-how, trade secrets, [and] clinical data.”

Also, even regarding US requirements for sensitizing substances reporting, the two sources quoted by Christian's answer seem pretty dubious (especially the latter--single author with no academic or industry affiliation, publishing in an obscure journal). The very same book from which the misleading red-underlined quote was taken says much more:

> The CFR’s biologics labeling regulations (21 CFR §610.61[l])
provide that “known sensitizing substances” should be listed
on the product label. Furthermore, 21 CFR §610.15(b) states
that “extraneous protein known to be capable of producing
allergenic effects in human subjects shall not be added to a
final virus medium of cell culture produced vaccines intended
for injection.” These regulations address the possibility that
animal-derived proteins present in the final formulation of a
vaccine can cause allergic reactions in some vaccine recipients.
(Other sensitizing substances, such as preservatives, are
addressed in other sections of these regulations.) Animal-
derived materials are used extensively in vaccine manufacturing, particularly in viral cultures. When viral vaccines are
grown in embryonated chicken eggs (influenza and yellow
fever vaccines) or chick embryo cell culture (measles or mumps
virus vaccines), the label will state that residual chicken proteins may be present in the final formulation (see Table 7.2).
Although hypersensitivity to any component of the vaccine is
a contraindication, the MMR-II and yellow fever vaccine (YF-
Vax) package inserts address the vaccination of persons with
hypersensitivity to eggs or egg protein.
>
> The two U.S.-licensed hepatitis B vaccines, Engerix-B (Glaxo-
SmithKline Biologicals) and Recombivax HB (Merck Sharpe &
Dohme Corp.), are recombinant DNA–derived proteins and
are produced in yeast; their package inserts note that residual
yeast protein may be present (Engerix-B contains not more
than 5% and Recombivax HB not more than 1% yeast protein).
Hepatitis B vaccine is contraindicated in persons with a history
of hypersensitivity to yeast; however, the ACIP has noted that
“no evidence exists that documents adverse reactions after
vaccination of persons with a history of yeast allergy.” The
HPV vaccines, Gardasil (Merck Sharp & Dohme Corp.) and
Gardasil 9 (Merck Sharp & Dohme Corp.), are also manufactured in yeast cells; according to the package inserts, both
vaccines contain less than 7 μg of yeast proteins per dose;
hypersensitivity to yeast is given as a contraindication to
receiving the vaccine. The other HPV vaccine, Cervarix (Glaxo-
SmithKline Biologicals), is manufactured in insect cells; levels
of insect cell proteins are reduced to less than 40 ng/dose.

There's a two-page table which follows, but I'm going to post just the part on influenza vaccines (as those are actually not discussed in the above quote, and while the table does a lot more detail on those included in the text above... the table-level of detail is on par with that for influenza ones.)

> [![enter image description here][2]][2]

One of the tree egg-grown vaccines in that table (Fluzone)
indeed doesn't seem to report ovalbumin contents, but the other two (Afluria, Fluarix) do report it. (I haven't investigated further why there is this discrepancy.)

As with other things related to medication approval (despite [some people's wishes](https://medicalsciences.stackexchange.com/questions/28829/what-is-the-authoritative-us-document-that-delineates-concrete-standards-for-vac)) there cannot be an entirely cookbook approach to these. It's clear from the above that despite the lack of strictly spelled out law/regulation requirements (like levels), the US regulators in conjunction with the manufacturers have provided sensible info on sensitizing substances in vaccines... the reporting of which is actually mandated in general terms in the federal regulations. 


  [1]: https://i.sstatic.net/DF8EB.png
  [2]: https://i.sstatic.net/o3Lkl.png