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, 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 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 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 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:
