Yes, your supplier will have an identical colour swatch to you provided they have purchased a SMART swatch and stored it correctly after purchase.
All Pantone fabric swatches come from the same source and are shipped globally to Pantone offices and authorised Pantone distributors. Swatches are sealed in a protective plastic wrap.
Any dyed cotton production for SMART swatches and the other cotton products occurs in one dye house and the quality control is very strict, colours have to match back to the original spectral data standard with a delta E measurement of 0.5 which is hardly noticeable when visual matching. Colour swatches also have to be on the same side of the standard in chroma, hue and lightness as all previous production of that colour.
Once purchased and opened it is recommend that you replace the swatch inside the plastic wrap and store in a cupboard or drawer away from light and dust to prevent fading and contamination.
If you are concerned about the standard being used by your supplier first check they are matching to fabric not paper. If they are matching to a Cotton Passport, Cotton Planner or Cotton Chip Set the colour could be affected by the background as these products have the fabric glued to card. They could also be using a swatch which is old and has been handled or left out in light.