Roughly speaking, the security parameter gives you an idea of how much time (or many operations) an attacker needs to break the scheme: that is, any algorithm that "breaks" the scheme has complexity at least $O(2^\lambda)$.
For example, on the end of section 5.2 of that paper, they conclude that successful attacks would take time $2^{\frac{\gamma}{\eta^2}}$. This is why, in section 3, they choose $\eta = \lambda^2$ and $\gamma = \lambda^5$, which means that the attack will take
$$2^{\frac{\gamma}{\eta^2}} = 2^{\frac{\lambda^5}{(\lambda^2)^2}} = 2^\lambda$$
Therefore, when we increase $\lambda$, the scheme becomes more secure. But of course, it has some impact on the scheme (otherwise, we could just pick $\lambda$ as large as we wanted and we would have an unbreakable cipher...). It means that other properties of the scheme must depend on $\lambda$ (for instance, the keys' size or the encryption time may increase when we choose bigger values for $\lambda$).
Almost all (modern) public-key schemes are presented in that way: with all the other parameters as functions of the security parameter $\lambda$. So, it is not exclusive to FHE schemes.
And even if a scheme is not presented in this way, we may reason in the same way the paper did in order to obtain a security parameter. For instance, let's pretend that the best attack against the RSA takes $2^{2\sqrt{n}}$. Then, to guarantee that attackers will take at least $2^\lambda$, we set $n = \frac{\lambda^2}{4}$. In practice, when we instantiate the scheme, we choose some value for $\lambda$, for instance, $128$. In this case, our $n$ should be $128^2 / 4 = 4096$. In other words, in this scenario, to guarantee a security of $128$ bits, we must choose the modulus size as $4096$ bits.
And about the comparisons... Well, you don't really want to compare the security parameters... You usually want to compare the keys' sizes and the timings (encryption, decryption, and key generation) to the same security parameter.