What is the criterion for genetic similarity among siblings and humans in general?
Your slight suspicion is close to correct. There are two different ways to discuss 'proportions of shared DNA' (identity by state, versus identity by descent). If two individuals share an allele that is identical by descent, it must by identical by state, but the reverse isn't true.
The 99% figure comes from comparing all of the base pairs and follows the concept of 'identity by state', where 'state' is A, T, C, or G. If individual 1 and individual 2 both have an A at position 39, they match. Average over all positions in the genome, and there's your 99% figure.
The second part of your suspicion is not quite correct, largely because of assortative mating and drift within local populations. The 50% figure comes from classical theory of Fisher, Haldane, and so on. They were trying to solve the mathematics starting from pedigrees and Punnett squares. They didn't have access to the high-throughput DNA sequence data that is common today, but you can follow a rare allele through a pedigree if it causes some observable outcome in its carriers. As a consequence, the classical theory is defined in terms of 'rare' alleles and follows the concept of 'identity by descent', and is tightly linked to the sorts of things you can observe in a pedigree. Having examined many pedigrees, you can generalise the patterns to derive classical population genetics theory. The key thing to remember here is that this branch of theory is expressed in terms of limit formulae: 'as variant A becomes rare, statement Y becomes closer to true'. In that context, the 50% figure is approximately the probability that your full-sibling has a rare allele, given that you have it.
That 50% is a direct consequence of the mechanics of meiosis: you have two copies of each allele. One copy of each allele was inherited from your mother, and the other from your father, at random. As a result, offspring have an exact 50% chance of inheriting any rare allele carried by one of their parents, and so are exactly 50% identical by descent to each of their parents. By the same reasoning, individuals are 50% identical by descent to their full-siblings, on average. In contrast, all humans are 99% identical by state, because a large portion of the genome is shared by all humans.