I would like to know when to use 同一 and when to use 一样 when describing items or people as similar? What is the difference in usage between these two phrases?
identicaland is usually used on identities.
equivalentand is usually used on properties or results.
我们去过同一个商店。We've been to the same store.
这两个商店卖的东西是一样的。These two stores sell the same stuffs.
同一首歌，不同的人唱，感觉是不一样的。The same song doesn't sound the same when it's sung by different singers.
Note there are some (idiomatic) exceptions to these rules, which can be another topic.
Since you've mentioned 'similar', neither of these two words means 'similar'. You may want to use
我有过类似的遭遇。I've had a similar experience.
我们的性格很相似. We're much alike in character.
几个目击者的说法都差不多. What the witnesses said are pretty much the same thing.
A little digression: adding to the complexity is the phrase “同样” So, I think the first step to distinguishing between 同一 , 同样, and 一样 would be breaking them down into characters.
meaning of '样' is “appearance, kind, type, sample”. as user58955 mentioned, things with the same appearance/kind/type/sample do not have to be the identical thing. (However, "同一" may not even point to the same "instance" of an object, see example 2 below)
“一” and “同” may both mean "same" or "similar". But "一" implies a constraint on number: there is only one same thing in question.
More examples that may be interesting and are harder to generalize:
There is a famous TV show called “同一首歌”, i.e. The Same Song. No matter how many times it is sung, it is still the same "song" in the sense of "the same combination of melody and lyrics, although it may be rendered with different instruments". "一样" is not used here.
我和他买了*同一*本书。我们买了*一样*的书。We bought the same book. We bought the same book(books). (Although s/he and I bought the "same book", we each have a copy. With "一样的书", s/he and I may have bought the same set of books, not just 1)
我和他上*同一*所大学，我和他*一样（同样）*厉害。 I went to the same university as he did. I am as successful as he is. （“我们上一样的大学” is ok, but may lead to being asked whether the universities are identical)
这两种塑料经过了*同一*道生产线，有着*一样（同样）*的品质。 These two types of plastics are processed in the same production line, and they are of similar qualities. (“一样的生产线” is also ok, but needs disambiguation. They may not be the identical production line.)
这两台iPhone 5看起来*一样（can't use 同样 here）*，但不是*同一*型号，因为一台是16GB的，一台是32GB的。 These two iPhone 5's look the same, but the are not of the identical model. The first is a 16GB model while the second one is a 32GB one. (“一样”'s usage as an adjective is demonstrated. It shall not be replaced with "同一")
在*同一*天，A市与B市天气*一样（同样）*糟糕。 City A's and City B's weathers are equally bad on the same day. （to use “同一” in the latter half of this sentence needs rephrasing a bit: “A市与B市有着同一种糟糕的天气” and sounds a little weird)
同一 is usually used to modify nouns to mean exactly the same object, as
the same species, not
同一物种 is okay),
the same room), etc.
一样 usually does not mean exactly the same object, we do not say 一样房间 to mean 'the same room'. It means, from the wording (一+样), both things look the same (but they are different objects). For instance,
一样的房间 means the decorations and furnishings are the same, but the rooms are different.