I am trying to understand the currency basis calculation and whether there is a difference in currency basis when quoted vs. OIS and -IBOR rates.
The curve Bloomberg EUR swaps curve (YCSW0045 Index) is indeed the euro equivalent of the Bloomberg USD swaps curve (YCSW0023 Index).
By equivalent I mean that each curves are constructed in the same manner : using sames types of instruments (deposits, FRAs, futures, swaps) with the same bootstrapping/implying method (exact fit vs best fit). For each you can also use OIS discounting or not.
DOCS + "bootstrapping" or "curve" in the search field will bring you Bloomberg's "Building the Bloomberg Interest Rate Curve – Definitions and Methodology" pdf, as of march 2016, as far as I remember. This document is quite thorough (as thorough as could a Bloomberg technical document be ...)
You can synthesise this with the single currency IBOR-OIS basis swap (SBS) in each currency.
For example paying the EUR/USD 10Y XCS @ -40bps, represents paying 3M Euribor -40 versus receiving 3M USD Libor flat. If you then buy a EUR 10Y OIS/IBOR SBS @ 8bps, this represents receiving 3M Euribor -8bps and paying EONIA flat. If you then sell a USD 10Y OIS/IBOR SBS @ 20bps, this represents paying 3M USD Libor - 20bps and receiving FFOIS flat.
Net you have synthesised paying a EUR/USD EONIA/FFOIS XCS @ -52bps (-40 +8 -20).
Note: due to the credit support annex (CSA) on the EUR SBS being EUR cash collateral through any clearing house this is different to that on the specific XCS instrument which (if it traded) would be USD collateralised on all aspects of the trade, so this will have a minor (probably negligible) impact.
Note: you can also use the same technique for other IBOR tenors such as 6M, where the standard XCS is 3M IBOR and there is a 3M/6M SBS market in each currency.