it could be, but it could also be that she's three, and lots of non gifted kids have similar sensitivities. it could also be that the sensitivities veer off the normal scale and into either spd or autistic spectrum disorders.
she sounds like a fun kid to be around though, and like lots of other bright children. you probably really won't know (or need to know) the 'cause' of her quirks for another few years yet, and if she still has them at 6 or 7 and doesn't outgrow them, then yes, it may be more than being a preschooler. 
either way, just enjoy her and carry on working on her social skills. it's just as important (and actually more so) for a gifted child to know how to behave in a social setting and bond with their peer group, so let her enjoy being three.
early testing will set you back around £500 at the cheapest, and isn't really advised unless you have a child who is exhibiting severe problems, as it isn't completely reliable so early. waiting until 7 or so is advised, but at school at the very least.
i think you should caryy on what you do normally at home, tbh. it seems to be working fine. teaching formal maths isn't required, you just incorporate the real world into your chat as you pass the time with her. the bright kids will just soak it all up like a sponge. lots of three year olds have already worked out a lot of simple maths just from climbing the stairs or whatever, tbh. so she might already know loads more than you realise. i had no idea that dd2 could read and describe graphs when she started school, or that ds1 could play shopping and work out multiplications to get to different numbers at 3.
i'm not sure who was more surprised, nursery or me. it didn't make any difference.
and ep testing is v funny. i soent most of the test going 'how do they know that?' kids are amazing.
ultimately though, remember she is three, and keep on jumping in puddles and going to the park, and playing on the slides, and squishing play doh into the carpet.
trampolines are always good value with busy kids, too. and they work out way cheaper than a gazillion bazillion classes. and if you put a couple of kids on together they make up games and stretch their imaginations that way, too.
please do make sure that the social side doesn't get lost in the realisation she may be bright.