I've noticed many of their runes look alike, so would that mean they share the same language? And if they did what's the Dwemer/Ayleid connection?

In general, the main merish language is Aldmeris, with all other mer languages being derivatives of Aldmeris. The exception is dunmeri, as they use daedric letters. As far as dialect, grammar rules, and spelling, [censored] if I know.
(waits for someone to point out how wrong I am)

ur rong

All spoken elven languages theoretically derive from Ehlnofex/Aldmeris. Ayleidic uses some of the same words as Dunmeri, but generally sounds different. The only examples of Dwemeris we have are really bizarre and out there, but theoretically they too are somehow derived from Aldmeris.

In terms of script, yes, they both use a similar alphabet (again, likely of Aldmeri origin). The Dwemer however have a second alphabet, however, which the Ayleid do not. You can probably write Dunmeri using an Aldmeri-derived alphabet, too, but their religious connection to the Daedra is why they favor a Daedric script.

Their languages are very simialer though, probably because of the Mer to be on mainland before the Altmer started moving, it was just Bosmer, Kajj't, Aylied, Dwe(b)mer, and Falmer. I'm sure they traded heavily, and the Khajjit were likely avoided.

It's like cuneiform. They all use variations of the same alphabet with different spoken languages.

All Elven languages are based on Aldmeris but there are differences. They're about as similar as the real world Romantic languages which all derived from Latin. To an outside listened you'll recognize many similar sounds, and the alphabet is the same, but they're different languages.