can i have multiple copies of a single pokemon?
Yes.  Eevee GX, Eevee V, and Eevee are different Pokémon, and you can have four of each in your deck. (Also, if they existed, Eevee EX, Eevee Vmax, Eevee prism star (1 copy), etc.). Most but not all of the ‘suffixes’ count as separate Pokémon.
That’s true for any Pokémon, so yes, Sylveon and Sylveon GX count separately. However, if the names are the same then they count as the same even if they’re different types or otherwise very different - Quagsire for example you can only have four total, even if they are fighting and water types individually.
Note that evolution also works that way - so something that evolves from Eevee cannot evolve from Eevee GX, unless it has an ability that says it can (as I think the GX does?).

Rules notes:  From the Compendium:

Q. Can you have 4 Electabuzz and 4 Electabuzz-EX in the same deck?

A. Yes, you can play 4 Electabuzz and 4 Electabuzz EX in the same deck, as they are considered different cards (similar to the ruling for Trainer Pokémon). (Sep 11, 2003 PUI Rules Team)


but

Q. Can you have 4 Electivire and 4 Electivire Lv.X in the same deck?

A. No, Lv.X Pokemon are considered the same as other pokemon with the same name. You may only have a maximum of 4 combined in your deck (i.e. 3 Electivire, 1 Electivire Lv.X). (Sep 25, 2008 PUI Rules Team)


GX and V and VMAX and Mega are all considered different, by the same rule as EX is.  Lv.X is not, for whatever reason that I can't entirely explain (I guess it's just the 'level' of the Pokémon?).