How to be sure services and researches offered by the University are not becoming cases of unfair competition?
I think your assumption that the university and private evaluators "offer the same service" is flawed. A private analysis will be delivered to a client and not to the general public. Being private, it may give some competitive advantage and so has monetary value. The university, on the other hand provides an information "service" to the general public so that all can benefit.
To "forbid" the university from doing some sort of research so that the possible results could be locked up for private rather than general benefit seems to be a pernicious idea.
And, the "service" of the university isn't actually free (as in beer). Resources go in to it that are paid in a variety of ways, sometimes from public monies since the public benefits. But the information it provides is free (as in speech).
But for your question about law, note that laws vary from place to place, and not all laws are wise.
EU law prohibits (or at least limits) states from giving businesses an unfair competitive advantage through subsidising them. These rules may come into play when universities offer services that compete with commercial players. This has been the subject of discussion in cases where universities would offer to do contract research for third parties, and would charge only for the the man hours and not for general overhead, allowing them to offer the research much cheaper than competitors.
I think (but I am not a lawyer) that this should not be an issue for the case at hand, since the university is offering the said service for free. I.e. they are not gaining any (direct) commercial benefit from offering the service. If the university were charging money for this service, it might be different.
Keep in mind that universities are typically either government-run institutions or non-profit organizations; in either case, their principal purpose is to benefit society in some way, rather than to earn a profit for their owners or shareholders. I think it's pretty well accepted for such organizations to provide goods and services for free or cheap, for the purposes of creating a social benefit, even if there are commercial businesses that also sell those goods or services. This is fundamentally different from typical "unfair competition" where a business sells goods at low cost for the purpose of eliminating the competition or getting customers locked in.
You can think of many analogies which most people find totally uncontroversial:
A local government funds a public library, where people can come and read books for free, with the goal of helping educate the population. Is this unfair competition with local bookstores?
A church opens a soup kitchen which gives away free meals, with the goal of helping the needy. Is this unfair competition with local restaurants?
The government operates and subsidizes a bus or metro system, funded mostly by taxes, whose fares don't cover the full costs of construction and operation. The goal is to help people get around, reduce traffic, and promote economic activity. Is this unfair competition with taxi companies?
Here, your university is providing expert analysis, presumably with the goal of increasing knowledge about art and preserving important cultural artworks. You aren't doing it to earn a profit. I don't see any problem.
Of course, this may depend in part on your political and philosophical views about how economies ought to operate, capitalism and socialism and free markets and so forth. But I think what I've said is pretty mainstream.
If you want to know whether your activities are legal, you should ask your university counsel rather than the Internet. But I'll be very surprised if there is any problem.