So, in most paintings done on the Catholic Monarchs (the Capitulation of Granada, Columbus' meeting etc...) there's this guy(s) accompanying them while wearing their royal standards as actual clothing and carrying a sceptre (something akin to the Lictors of the Roman Republic). What were they called and what was their function ?
That guy is a macero, and what he carries is not a scepter but a mace. The surcoat with the arms of the monarchs is called a tabard, and was something used by maceros, heralds, and kings of arms as a way of identification of office.
As you mention, the macers are something akin to the Roman lictors, being not only people protecting the monarchs, but also a very clear kingly prerogative, as it is very much attested in the Ordinances emanated from the Cortes gathered in Toledo in 1480, where we can read the following in article 119:
By the aforementioned sollicitors of our realms another supplication was made to us, saying the We knew well that the kings of these realms, in respect to their royal dignities, some pre-eminences, insignia, and ceremonies were owed as they were instituted in great secular dignities, such as having macers, and raised sword before them, and to put a coronet on their royal arms [...]
Macers can even be seen today, but they no longer have a duty of protection (there are municipal police forces, the national police, the autonomic polices, or the civil guard for that), they are ceremonial. You can see them in different town halls, autonomic parliaments, or even the Congreso de los Diputados.
For actual protection, the monarchs actually had a guard, the corps of monteros de Espinosa. As time went on, different corps appeared, like the Guards of Castile, the Yellow Guard, the Purple Guard, the Walloon Guard, and finally the Royal Guard.