J.J.
hi C.! this sounds totally normal to me and very common and there's probably nothing you can do! :) no, it's not stranger anxiety if there's no stranger. it's mommy-love! and it's really normal for 6-12 months.
my son was totally fine w strangers until 11 months and then he became very clingy; he stopped being so physically clingy once he could walk and do other things but now that he's 2.5 and we have a 9 mo old girl he's on top of me all the time. he will hang on my leg as i drag him down the hall. meanwhile, my daughter would go to anybody until about 6 weeks ago, now she's not happy about strangers at all. she's still friendly and independent, she is just very decisive about who she wants to be with and when. she will visit w strangers or even people she knows but doesn't like to be picked up. and late in the afternoon she will insist on being carried around quite a lot for "no reason," except this;
the reason is they love us!
you can't blame them. we're the whole world to them. they need and love to be close w us. i think you should enjoy it because before you know it he'll be 4 or 5 and off to school all day and you'll miss him like crazy. and then he'll be 12 and he'll mouth off to you and you will long for the days when he wanted you close to him all the time. i know i will.
the other thing to remember i think is that the more independent they become they also become more clingy at other times; it's a rubberband effect. they need to be bolstered by you to be able to take chances and grow. and we should give them all the love we can. especially because as they get older it gets harder and you'll have days where you're just exhausted and you can't do everything. so if i was you i would give love while the giving is easy.
good luck!