Can our chemical elements be differents on other universes?
I admit this question is far out of my depth, but it's interesting enough that I feel I should give it a shot.

TL;DR - The question is so hard as to be almost unanswerable in detail. Things can be very different, though not arbitrarily different, and the exact line between the two is very difficult to define. There have been some mild investigations on the chemistry of toy universes, however, one of which I describe below.

The problem with thinking about other universes is that there are potentially a lot of different universes which can arise. Famously, string theory currently posits that there are something like $10^{500}$ "valid" universes, each with different details regarding physical laws and physical constants. Furthermore, universes are very complex objects in the sense that there is a ton of interrelation between everything, so even minor changes to details can create wildly different results. Not just that, but there apparently are many parameters to fiddle with, which would create wildly different periodic tables, often apparently breaking any possibility of life. This all leads to fine-tuning problems.

For example, if the attraction between two protons were about 2% larger, then every proton in the Universe would be bound in the form of a diproton nucleus, i.e. the nucleus of helium-2 ($\ce{^2He}$, which is just barely unstable in our Universe), meaning the entire periodic table would collapse to a single element. Apparently there's no very interesting chemistry in that universe! Now, what exactly would have to be tweaked in order to get a 2% stronger attraction between protons while not breaking everything else, I don't know.

So it seems thinking about chemistry in different universes is far too complicated to approach in almost any situation, but still there are some investigations. One particular case I have come across is what would happen in a universe without weak interactions.

Most of Chemistry ultimately has to do with studying different aspects of electromagnetic interactions. However, the stability of atomic nuclei depends immensely on strong interactions (aka strong nuclear force), and to a lesser extent on weak interactions (aka weak nuclear force). The authors in the article linked above analyzed the effects of taking a universe similar to ours, but removing the weak interaction. There are several differences with respect to the evolution of the universe (big bang nucleosynthesis, stellar nucleosynthesis, etc) and the abundance of the elements, but thinking just about their stability, it seems that in such a universe, a large portion of the periodic table would still exist.

Additionally, given that the weak interaction mediates the decay of neutrons in our Universe, its absence means neutrons are completely stable subatomic particles (in our Universe free neutrons decay with a half-life of around 8 minutes). This also means that neutron-rich nuclei are more stable to decay in the hypothetical weakless universe. In our Universe, deuterium ($\ce{^2_1H}$)is the most neutron-rich stable hydrogen isotope, while tritium ($\ce{^3_1H}$) is stable enough, with a half-life of about 11 years, to be chemically important (heavier hydrogen isotopes can essentially be neglected). However, in the weakless universe, it seems that even hydrogen nuclei as rich as $\ce{^6_1H}$ are indefinitely stable, and there are possibly even more neutron-rich isotopes. See the following figure:



(Source)

For comparison purposes, the most neutron-rich stable isotopes for each element in our Universe are $\ce{^2_1H}$, $\ce{^4_2He}$, $\ce{^7_3Li}$, $\ce{^9_4Be}$, $\ce{^11_5B}$, $\ce{^13_6C}$, $\ce{^15_7N}$ and $\ce{^18_8O}$.

Due to this added nuclear stability, the periodic table in the hypothetical weakless universe could actually have more stable elements in it. In our Universe, elements beyond about the 110th are quite unstable, being produced naturally in small amounts, if any, and they will not likely have great significance in Chemistry. However, highly neutron-rich nuclei could make ultraheavy elements more accessible, even in natural conditions such as supernovae.